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5048 a Ft(])i(implies)440 5147 y(an)i(unconditional)c(b)r(ound)k(of)g Fr(O)r Ft(\()p Fr(k)1575 5085 y Fk(p)p 1645 5085 167 4 v 1645 5147 a Ft(log)13 b Fr(k)s Ft(\001\).)41 b(After)30 b(submission)25 b(of)k(this)f(man)n(uscript,)440 5247 y(breakthroughs)c(of)h(Norin,)g(P)n(ostle)e(and)j(Song)f([)1963 5247 y SDict begin H.S end 1963 5247 a 0 TeXcolorgray Ft(13)p 0 TeXcolorgray 2046 5193 a SDict begin H.R end 2046 5193 a 2046 5247 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.NPS20) cvn H.B /ANN pdfmark end 2046 5247 a Ft(,)2094 5247 y SDict begin H.S end 2094 5247 a 0 TeXcolorgray Ft(12)p 0 TeXcolorgray 2177 5193 a SDict begin H.R end 2177 5193 a 2177 5247 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.P20) cvn H.B /ANN pdfmark end 2177 5247 a Ft(])h(ha)n(v)n(e)f (yielded)e(an)j(unconditional)440 5347 y(b)r(ound)f(of)f Fr(O)r Ft(\()p Fr(k)s Ft(\(log)14 b Fr(k)s Ft(\))1159 5316 y Fh(\014)1204 5347 y Ft(\))25 b(for)f(an)n(y)g Fr(\014)j(>)c Ft(4,)i(later)d(impro)n(v)n(ed)g(further)i(to)g Fr(O)r Ft(\()p Fr(k)s Ft(\(log)14 b(log)g Fr(k)s Ft(\))3212 5316 y Fp(6)3249 5347 y Ft(\).)p 0 TeXcolorgray 1851 5662 a(3)p 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 440 133 a SDict begin H.S end 440 133 a 440 133 a SDict begin H.R end 440 133 a 440 133 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 440 133 a 0 TeXcolorgray 0 TeXcolorgray 565 531 a Ft(Imp)r(ortan)n(t)24 b(to)h(the)h(pro)r(of)f(of)h(Theorem)1859 531 y SDict begin H.S end 1859 531 a 0 TeXcolorgray Ft(7)p 0 TeXcolorgray 1900 478 a SDict begin H.R end 1900 478 a 1900 531 a 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SDict begin H.R end 772 876 a 772 930 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 772 930 a 28 w Ft(\(see)f(Conjecture)1389 930 y SDict begin H.S end 1389 930 a 0 TeXcolorgray Ft(15)p 0 TeXcolorgray 1472 876 a SDict begin H.R end 1472 876 a 1472 930 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.15) cvn H.B /ANN pdfmark end 1472 930 a 27 w Ft(b)r(elo)n(w\).)565 1029 y(F)-7 b(or)32 b Fr(k)j Ft(=)d(5,)i(the)g(construction)d(in)i (Section)2022 1029 y SDict begin H.S end 2022 1029 a 0 TeXcolorgray Ft(3)p 0 TeXcolorgray 2063 976 a SDict begin H.R end 2063 976 a 2063 1029 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 2063 1029 a 33 w Ft(is)f(\\v)n(ery)g(lo)r(cal",)f(in)i(the)g(sense)g(that) 440 1129 y Fs(every)e Ft(t)n(w)n(o)23 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b(\(Note)28 b(that)f Fr(k)22 b Fk(\000)1695 1395 y Fp(1)p 1695 1409 34 4 v 1695 1457 a(2)1761 1428 y Fr(<)1858 1395 y Fp(3)p 1858 1409 V 1858 1457 a(2)1902 1428 y Ft(\()p Fr(k)f Fk(\000)d Ft(2\))28 b(if)f(and)g(only)f(if)h Fr(k)f(>)d Ft(5.\))440 1594 y SDict begin H.S end 440 1594 a 440 1594 a SDict begin 12 H.A end 440 1594 a 440 1594 a SDict begin [/View [/XYZ H.V]/Dest (thm.8) cvn /DEST pdfmark end 440 1594 a 0 TeXcolorgray Fl(Theorem)40 b(8.)p 0 TeXcolorgray 46 w Fs(L)l(et)d Fr(k)i Fk(\025)e Ft(5)p Fs(.)60 b(F)-6 b(or)38 b(any)g Fr(K)1839 1606 y Fh(k)1879 1594 y Fs(-minor-fr)l(e)l(e)g(multigr)l(aph)g Fr(G)f Fs(of)i(maximum)440 1694 y(de)l(gr)l(e)l(e)30 b Ft(\001)g Fs(such)f(that)h(every)g(two)g(of)g(its)g(e)l(dges)g(ar)l(e)g(incident) g(or)g(joine)l(d)h(by)f(an)f(e)l(dge,)i(the)440 1793 y(str)l(ong)e(clique)i(numb)l(er)e(of)h Fr(G)g Fs(satis\014es)g Fr(!)1780 1763 y Fq(0)1777 1814 y Fp(2)1814 1793 y Ft(\()p Fr(G)p Ft(\))24 b Fk(\024)f Ft(\()p Fr(k)e Fk(\000)2244 1761 y Fp(1)p 2244 1775 34 4 v 2244 1822 a(2)2287 1793 y Ft(\)\001)p Fr(:)440 1959 y Ft(A)28 b Fs(blown-up)i Ft(5)p Fs(-cycle)k Ft(of)28 b(order)e Fr(t)i Ft(is)e(the)j(graph)d (obtained)g(from)h(the)h(5-cycle)d(b)n(y)j(substi-)440 2059 y(tuting)c(eac)n(h)g(v)n(ertex)f Fr(u)i Ft(with)f(a)g(stable)f (set)i Fr(S)1858 2071 y Fh(u)1926 2059 y Ft(of)f(size)f Fr(t)i Ft(and)g(substituting)e(eac)n(h)g(edge)i Fr(uv)440 2159 y Ft(with)30 b(a)f(complete)g(bipartite)f(graph)h(b)r(et)n(w)n (een)h Fr(S)2013 2171 y Fh(u)2087 2159 y Ft(and)g Fr(S)2302 2171 y Fh(v)2341 2159 y Ft(.)45 b(Note)30 b(that)g(for)g(o)r(dd)g Fr(t)e Fk(\025)f Ft(3,)440 2258 y(the)j(blo)n(wn-up)f(5-cycle)e(of)j (order)f Fr(t)g Ft(has)h(no)f Fr(K)1951 2256 y Fg(5)p 1951 2265 29 3 v 1951 2299 a(2)1989 2279 y Fh(t)p Fp(+)2075 2256 y Fg(1)p 2075 2265 V 2075 2299 a(2)2117 2258 y Ft(-minor)e(\(for)j (otherwise,)f(b)n(y)g(the)h(pi-)440 2358 y(geonhole)c(principle,)g(at)j (least)e(one)h(branc)n(h)g(set)g(of)h(suc)n(h)f(a)g(minor)e(w)n(ould)h 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0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3) cvn H.B /ANN pdfmark end 2275 2657 a Ft(\))h(is)e(exact)g(for)h(in\014nitely)e(man)n(y)440 2756 y(pairs)i(\()p Fr(k)s(;)14 b Ft(\001\).)38 b(By)28 b(a)f(more)f(in)n(v)n(olv)n(ed)f(m)n(ultigraph)f(construction)i(w)n(e)i (giv)n(e)e(in)h(Section)3240 2756 y SDict begin H.S end 3240 2756 a 0 TeXcolorgray Ft(7)p 0 TeXcolorgray 3281 2703 a SDict begin H.R end 3281 2703 a 3281 2756 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.7) cvn H.B /ANN pdfmark end 3281 2756 a Ft(,)440 2856 y(for)k(eac)n(h)g Fr(k)i Fk(\025)d Ft(5)h(and)h(\001)e Fk(\025)g Fr(k)24 b Fk(\000)d Ft(2,)32 b(Theorem)1971 2856 y SDict begin H.S end 1971 2856 a 0 TeXcolorgray Ft(8)p 0 TeXcolorgray 2012 2802 a SDict begin H.R end 2012 2802 a 2012 2856 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.8) cvn H.B /ANN pdfmark end 2012 2856 a 32 w Ft(is)f(sharp)g(up)h(to)g(an)f(additiv)n(e)f(term)g (of)440 2956 y(order)h(at)g(most)g(quadratic)f(in)h Fr(k)s Ft(.)49 b(F)-7 b(or)32 b Fr(k)h Fk(\024)c Ft(4,)k(the)f(b)r(ound)h (still)c(holds,)i(but)i(then)f(it)f(is)440 3055 y(not)d(necessarily)c (\(asymptotically\))f(sharp,)j(e.g.)i(Theorem)2377 3055 y SDict begin H.S end 2377 3055 a 0 TeXcolorgray Ft(6)p 0 TeXcolorgray 2419 3002 a SDict begin H.R end 2419 3002 a 2419 3055 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.6) cvn H.B /ANN pdfmark end 2419 3055 a Ft(.)565 3155 y(Returning)22 b(to)h(just)h(simple)d(graphs,)i(note)g(that)g(a)g (minor)e(adaptation)h(of)h(Theorem)3263 3155 y SDict begin H.S end 3263 3155 a 0 TeXcolorgray Ft(2)p 0 TeXcolorgray 3304 3101 a SDict begin H.R end 3304 3101 a 3304 3155 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2) cvn H.B /ANN pdfmark end 3304 3155 a 440 3254 a Ft(\(see)28 b(Theorem)957 3254 y SDict begin H.S end 957 3254 a 0 TeXcolorgray Ft(11)p 0 TeXcolorgray 1040 3201 a SDict begin H.R end 1040 3201 a 1040 3254 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.11) cvn H.B /ANN pdfmark end 1040 3254 a 28 w Ft(b)r(elo)n(w\))f(yields)e(the)j(follo)n(wing)23 b(w)n(eak)n(er)j(v)n(ersion)f(of)j(Conjecture)3136 3254 y SDict begin H.S end 3136 3254 a 0 TeXcolorgray Ft(3)p 0 TeXcolorgray 3177 3201 a SDict begin H.R end 3177 3201 a 3177 3254 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3) cvn H.B /ANN pdfmark end 3177 3254 a Ft(.)440 3421 y SDict begin H.S end 440 3421 a 440 3421 a SDict begin 12 H.A end 440 3421 a 440 3421 a SDict begin [/View [/XYZ H.V]/Dest (thm.9) cvn /DEST pdfmark end 440 3421 a 0 TeXcolorgray Fl(Theorem)k(9.)p 0 TeXcolorgray 41 w Fs(F)-6 b(or)31 b(any)g Fr(K)1350 3433 y Fh(k)1391 3421 y Fs(-minor-fr)l(e)l(e)g(\(simple\))g(gr)l(aph)h Fr(G)f Fs(of)h(maximum)e(de)l(gr)l(e)l(e)i Ft(\001)p Fs(,)440 3520 y(the)e(str)l(ong)f(clique)i(numb)l(er)e(of)h Fr(G)g Fs(satis\014es)g Fr(!)1918 3490 y Fq(0)1915 3541 y Fp(2)1952 3520 y Ft(\()p Fr(G)p Ft(\))24 b Fk(\024)f Ft(\()p Fr(k)e Fk(\000)d Ft(1\)\(\001)h(+)f(1\))p Fs(.)440 3686 y Ft(Although)23 b(at)g(the)h(end)f(of)h(the)f(pap)r(er)g(w)n(e)g (discuss)f(ho)n(w)h(to)g(sharp)r(en)g(this)g(b)r(ound)h(sligh)n(tly)-7 b(,)440 3786 y(already)23 b(the)j(blo)n(wn-up)e(5-cycle)f(describ)r(ed) h(just)i(ab)r(o)n(v)n(e)e(sho)n(ws)g(the)i(b)r(ound)g(to)f(b)r(e)h (sharp)440 3885 y(up)k(to)h(a)e Fr(O)r Ft(\(\001\))j(or)d Fr(O)r Ft(\()p Fr(k)s Ft(\))i(additiv)n(e)d(factor)i(when)g(\001)d(=)g (2)p Fr(t)g Ft(=)2448 3853 y Fp(4)p 2448 3867 34 4 v 2448 3914 a(5)2491 3885 y Ft(\()p Fr(k)c Fk(\000)2684 3853 y Fp(1)p 2684 3867 V 2684 3914 a(2)2727 3885 y Ft(\))30 b(for)g(o)r(dd)g Fr(t)p Ft(.)45 b(F)-7 b(or)440 3985 y(relativ)n(ely)17 b(larger)h(\001,)23 b(sp)r(eci\014cally)18 b(for)i(\001)j Fk(\025)g Fr(k)8 b Fk(\000)e Ft(2,)22 b(a)e(complete)g(bipartite)f(graph)h Fr(K)3105 3997 y Fh(k)q Fq(\000)p Fp(2)p Fh(;)p Fp(\001)440 4085 y Ft(ha)n(ving)30 b(parts)i(of)h(size)e Fr(k)24 b Fk(\000)d Ft(2)32 b(and)h(\001,)h(to)e (one)g(v)n(ertex)g(in)f(the)i(larger)d(part)i(of)g(whic)n(h)f(is)440 4184 y(attac)n(hed)23 b(\001)11 b Fk(\000)g Fr(k)i Ft(+)e(2)22 b(p)r(endan)n(t)i(edges,)g(is)f(a)g Fr(K)1904 4196 y Fh(k)1944 4184 y Ft(-minor-free)e(graph)h(of)i(maxim)n(um)19 b(degree)440 4284 y(\001)28 b(with)f(strong)f(clique)g(n)n(um)n(b)r(er) g(\()p Fr(k)c Fk(\000)c Ft(1\)\(\001)h Fk(\000)f Ft(1\))g(+)g(1.)565 4384 y(By)g(another)g(adaptation)f(of)h(Theorem)1820 4384 y SDict begin H.S end 1820 4384 a 0 TeXcolorgray Ft(2)p 0 TeXcolorgray 1862 4330 a SDict begin H.R end 1862 4330 a 1862 4384 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2) cvn H.B /ANN pdfmark end 1862 4384 a Ft(,)i(w)n(e)f(also)d (ha)n(v)n(e)i(the)h(follo)n(wing)c(as)j(a)g(corollary)440 4483 y(of)23 b(Reed)g(and)f(Seymour's)f(result)g([)1534 4483 y SDict begin H.S end 1534 4483 a 0 TeXcolorgray Ft(14)p 0 TeXcolorgray 1618 4430 a SDict begin H.R end 1618 4430 a 1618 4483 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ReSe98) cvn H.B /ANN pdfmark end 1618 4483 a Ft(])h(that)h(ev)n(ery) f Fr(K)2124 4495 y Fh(k)2164 4483 y Ft(-minor-free)e(graph)h(has)h (fractional)440 4583 y(c)n(hromatic)37 b(n)n(um)n(b)r(er)h(at)i(most)f (2\()p Fr(k)29 b Fk(\000)d Ft(1\).)74 b(The)39 b Fs(fr)l(actional)k (str)l(ong)e(chr)l(omatic)h(index)440 4682 y Fr(\037)492 4652 y Fq(0)492 4706 y Fp(2)p Fh(;f)588 4682 y Ft(\()p Fr(G)p Ft(\))28 b(of)g(a)f(graph)g Fr(G)g Ft(is)g Fr(\037)1373 4694 y Fh(f)1416 4682 y Ft(\()p Fr(L)p Ft(\()p Fr(G)p Ft(\))1634 4652 y Fp(2)1672 4682 y Ft(\).)37 b(Note)28 b Fr(!)2020 4652 y Fq(0)2017 4703 y Fp(2)2054 4682 y Ft(\()p Fr(G)p Ft(\))c Fk(\024)e Fr(\037)2346 4652 y Fq(0)2346 4706 y Fp(2)p Fh(;f)2442 4682 y Ft(\()p Fr(G)p Ft(\))i Fk(\024)f Fr(\037)2735 4652 y Fq(0)2735 4703 y Fp(2)2772 4682 y Ft(\()p Fr(G)p Ft(\))28 b(alw)n(a)n(ys.)440 4848 y SDict begin H.S end 440 4848 a 440 4848 a SDict begin 12 H.A end 440 4848 a 440 4848 a SDict begin [/View [/XYZ H.V]/Dest (thm.10) cvn /DEST pdfmark end 440 4848 a 0 TeXcolorgray Fl(Theorem)e(10.)p 0 TeXcolorgray 38 w Fs(F)-6 b(or)26 b(any)h Fr(K)1380 4860 y Fh(k)1421 4848 y Fs(-minor-fr)l(e)l(e)f(\(simple\))i(gr)l(aph)g Fr(G)e Fs(of)i(maximum)e(de)l(gr)l(e)l(e)h Ft(\001)p Fs(,)440 4948 y(the)j(fr)l(actional)h(str)l(ong)f(chr)l(omatic)h(index) f(of)g Fr(G)g Fs(satis\014es)g Fr(\037)2362 4918 y Fq(0)2362 4972 y Fp(2)p Fh(;f)2458 4948 y Ft(\()p Fr(G)p Ft(\))24 b Fk(\024)e Ft(2\()p Fr(k)g Fk(\000)c Ft(1\)\(\001)g(+)g(1\))p Fs(.)440 5114 y Ft(In)28 b(fact,)f(impro)n(ving)d(on)j(the)h(factor)e (2)h(in)g(Theorem)2135 5114 y SDict begin H.S end 2135 5114 a 0 TeXcolorgray Ft(10)p 0 TeXcolorgray 2218 5061 a SDict begin H.R end 2218 5061 a 2218 5114 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.10) cvn H.B /ANN pdfmark end 2218 5114 a 27 w Ft(is)g(equiv)-5 b(alen)n(t)25 b(to)i(impro)n(ving)d (on)440 5214 y(the)j(factor)f(2)h(in)f(Reed)h(and)g(Seymour's)e(result) g(\(see)i(Corollary)2515 5214 y SDict begin H.S end 2515 5214 a 0 TeXcolorgray Ft(12)p 0 TeXcolorgray 2598 5160 a SDict begin H.R end 2598 5160 a 2598 5214 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.12) cvn H.B /ANN pdfmark end 2598 5214 a Ft(\).)37 b(It)27 b(is)f(natural)f(here)440 5313 y(for)j(us)g(to)f(reiterate)f(the)j(imp)r(ortance)c(of)j(Theorem) 2136 5313 y SDict begin H.S end 2136 5313 a 0 TeXcolorgray Ft(2)p 0 TeXcolorgray 2177 5260 a SDict begin H.R end 2177 5260 a 2177 5313 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest 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1482 y(with)36 b(underlying)e(simple)g(subgraph)i(consisting)e(of)j(a)f (disjoin)n(t)f(union)g(of)i(o)r(dd)g(cycles)440 1581 y(and)f(edges.)63 b(In)36 b(Section)1302 1581 y SDict begin H.S end 1302 1581 a 0 TeXcolorgray Ft(6)p 0 TeXcolorgray 1344 1528 a SDict begin H.R end 1344 1528 a 1344 1581 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.6) cvn H.B /ANN pdfmark end 1344 1581 a 36 w Ft(w)n(e)g(pro)n(v)n(e)f (Theorem)2104 1581 y SDict begin H.S end 2104 1581 a 0 TeXcolorgray Ft(6)p 0 TeXcolorgray 2145 1528 a SDict begin H.R end 2145 1528 a 2145 1581 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.6) cvn H.B /ANN pdfmark end 2145 1581 a 37 w Ft(on)h Fr(K)2377 1593 y Fp(4)2413 1581 y Ft(-minor-free)d(m)n(ultigraphs.)440 1681 y(In)c(Section)836 1681 y SDict begin H.S end 836 1681 a 0 TeXcolorgray Ft(7)p 0 TeXcolorgray 877 1627 a SDict begin H.R end 877 1627 a 877 1681 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.7) cvn H.B /ANN pdfmark end 877 1681 a Ft(,)g(w)n(e)f(pro)n(v)n(e) f(Theorem)1628 1681 y SDict begin H.S end 1628 1681 a 0 TeXcolorgray Ft(8)p 0 TeXcolorgray 1670 1627 a SDict begin H.R end 1670 1627 a 1670 1681 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.8) cvn H.B /ANN pdfmark end 1670 1681 a 28 w Ft(and)h(w)n(e)g(construct)g(a)g(m)n(ultigraph)c (certifying)i(its)440 1780 y(asymptotic)j(sharpness.)46 b(W)-7 b(e)31 b(also)e(pro)n(v)n(e)g(v)-5 b(arian)n(ts)29 b(of)i(Theorems)2635 1780 y SDict begin H.S end 2635 1780 a 0 TeXcolorgray Ft(7)p 0 TeXcolorgray 2677 1727 a SDict begin H.R end 2677 1727 a 2677 1780 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.7) cvn H.B /ANN pdfmark end 2677 1780 a 31 w Ft(and)2873 1780 y SDict begin H.S end 2873 1780 a 0 TeXcolorgray Ft(8)p 0 TeXcolorgray 2914 1727 a SDict begin H.R end 2914 1727 a 2914 1780 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.8) cvn H.B /ANN pdfmark end 2914 1780 a 31 w Ft(where)g(the)440 1880 y(condition)26 b(of)i Fr(K)971 1892 y Fh(k)1012 1880 y Ft(-minor-freeness)c(is)j (relaxed)f(to)j(the)f(absence)g(of)g(one)g(particular)d(t)n(yp)r(e)440 1980 y(of)37 b Fr(K)615 1992 y Fh(k)655 1980 y Ft(-minor.)61 b(A)n(t)37 b(the)g(end,)i(w)n(e)d(revisit)e(strong)h(cliques)g(in)g (simple)f(graphs)h(and)i(w)n(e)440 2079 y(discuss)26 b(\(nonasymptotic\))g(strengthenings)f(of)j(Theorem)2344 2079 y SDict begin H.S end 2344 2079 a 0 TeXcolorgray Ft(9)p 0 TeXcolorgray 2385 2026 a SDict begin H.R end 2385 2026 a 2385 2079 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.9) cvn H.B /ANN pdfmark end 2385 2079 a Ft(.)440 2225 y SDict begin H.S end 440 2225 a 440 2225 a SDict begin 12 H.A end 440 2225 a 440 2225 a SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark end 440 2225 a 129 x Fu(2)135 b(Simple)42 b(reductions)440 2536 y Ft(W)-7 b(e)34 b(pro)n(v)n(e)e(an)i(omnibus)e(generalisation)c(of)34 b(Theorems)2303 2536 y SDict begin H.S end 2303 2536 a 0 TeXcolorgray Ft(2)p 0 TeXcolorgray 2344 2482 a SDict begin H.R end 2344 2482 a 2344 2536 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2) cvn H.B /ANN pdfmark end 2344 2536 a Ft(,)2401 2536 y SDict begin H.S end 2401 2536 a 0 TeXcolorgray Ft(9)p 0 TeXcolorgray 2443 2482 a SDict begin H.R end 2443 2482 a 2443 2536 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.9) cvn H.B /ANN pdfmark end 2443 2536 a 34 w Ft(and)2644 2536 y SDict begin H.S end 2644 2536 a 0 TeXcolorgray Ft(10)p 0 TeXcolorgray 2727 2482 a SDict begin H.R end 2727 2482 a 2727 2536 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.10) cvn H.B /ANN pdfmark end 2727 2536 a 34 w Ft(in)f(Theorem)3221 2536 y SDict begin H.S end 3221 2536 a 0 TeXcolorgray Ft(11)p 0 TeXcolorgray 3304 2482 a SDict begin H.R end 3304 2482 a 3304 2536 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.11) cvn H.B /ANN pdfmark end 3304 2536 a 440 2635 a Ft(b)r(elo)n(w.)53 b(Note)34 b(that)f(Theorem)1474 2635 y SDict begin H.S end 1474 2635 a 0 TeXcolorgray Ft(2)p 0 TeXcolorgray 1516 2582 a SDict begin H.R end 1516 2582 a 1516 2635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2) cvn H.B /ANN pdfmark end 1516 2635 a 33 w Ft(follo)n(ws)e(from)h(parts)2249 2656 y SDict begin H.S end 2249 2656 a 0 TeXcolorgray -21 x Ft(\()p Fr(v)s Ft(\))p 0 TeXcolorgray 2356 2573 a SDict begin H.R end 2356 2573 a 2356 2635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.5) cvn H.B /ANN pdfmark end 2356 2635 a 34 w Ft(and)2557 2656 y SDict begin H.S end 2557 2656 a 0 TeXcolorgray -21 x Ft(\()p Fr(iii)p Ft(\))p 0 TeXcolorgray 2708 2573 a SDict begin H.R end 2708 2573 a 2708 2635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.3) cvn H.B /ANN pdfmark end 2708 2635 a 33 w Ft(of)i(Theorem)3198 2635 y SDict begin H.S end 3198 2635 a 0 TeXcolorgray Ft(11)p 0 TeXcolorgray 3281 2582 a SDict begin H.R end 3281 2582 a 3281 2635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.11) cvn H.B /ANN pdfmark end 3281 2635 a Ft(.)440 2735 y(P)n(art)634 2756 y SDict begin H.S end 634 2756 a 0 TeXcolorgray -21 x Ft(\()p Fr(i)p Ft(\))p 0 TeXcolorgray 728 2673 a SDict begin H.R end 728 2673 a 728 2735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.1) cvn H.B /ANN pdfmark end 728 2735 a 34 w Ft(of)f(Theorem)1220 2735 y SDict begin H.S end 1220 2735 a 0 TeXcolorgray Ft(11)p 0 TeXcolorgray 1303 2681 a SDict begin H.R end 1303 2681 a 1303 2735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.11) cvn H.B /ANN pdfmark end 1303 2735 a 34 w Ft(implies)d(Theorem)1982 2735 y SDict begin H.S end 1982 2735 a 0 TeXcolorgray Ft(9)p 0 TeXcolorgray 2023 2681 a SDict begin H.R end 2023 2681 a 2023 2735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.9) cvn H.B /ANN pdfmark end 2023 2735 a Ft(,)36 b(while)c(part)2491 2756 y SDict begin H.S end 2491 2756 a 0 TeXcolorgray -21 x Ft(\()p Fr(iv)s Ft(\))p 0 TeXcolorgray 2628 2673 a SDict begin H.R end 2628 2673 a 2628 2735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.4) cvn H.B /ANN pdfmark end 2628 2735 a 34 w 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0.79701 pgfw 1 0 0 TeXcolorrgb 24.55061 -14.17426 moveto 0.0 28.34853 lineto -24.55061 -14.17426 lineto closepath pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 25.95952 -14.17426 moveto 25.95952 -13.39615 25.32872 -12.76535 24.55061 -12.76535 curveto 23.77249 -12.76535 23.14171 -13.39615 23.14171 -14.17426 curveto 23.14171 -14.95236 23.77249 -15.58316 24.55061 -15.58316 curveto 25.32872 -15.58316 25.95952 -14.95236 25.95952 -14.17426 curveto closepath 24.55061 -14.17426 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 24.55061 -14.17426 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 1.40889 28.34853 moveto 1.40889 29.12663 0.7781 29.75743 0.0 29.75743 curveto -0.7781 29.75743 -1.40889 29.12663 -1.40889 28.34853 curveto -1.40889 27.5704 -0.7781 26.93962 0.0 26.93962 curveto 0.7781 26.93962 1.40889 27.5704 1.40889 28.34853 curveto closepath 0.0 28.34853 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 0.0 28.34853 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -23.14171 -14.17426 moveto -23.14171 -13.39615 -23.77249 -12.76535 -24.55061 -12.76535 curveto -25.32872 -12.76535 -25.95952 -13.39615 -25.95952 -14.17426 curveto -25.95952 -14.95236 -25.32872 -15.58316 -24.55061 -15.58316 curveto -23.77249 -15.58316 -23.14171 -14.95236 -23.14171 -14.17426 curveto closepath -24.55061 -14.17426 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 -24.55061 -14.17426 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore 0 TeXcolorgray restore save 0.79701 pgfw 1 0 0 TeXcolorrgb 29.46072 -17.00911 moveto 0.0 34.01823 lineto -29.46072 -17.00911 lineto closepath pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 30.86963 -17.00911 moveto 30.86963 -16.23099 30.23885 -15.6002 29.46072 -15.6002 curveto 28.68262 -15.6002 28.05182 -16.23099 28.05182 -17.00911 curveto 28.05182 -17.78722 28.68262 -18.41801 29.46072 -18.41801 curveto 30.23885 -18.41801 30.86963 -17.78722 30.86963 -17.00911 curveto closepath 29.46072 -17.00911 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 29.46072 -17.00911 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 1.40889 34.01823 moveto 1.40889 34.79634 0.7781 35.42714 0.0 35.42714 curveto -0.7781 35.42714 -1.40889 34.79634 -1.40889 34.01823 curveto -1.40889 33.24013 -0.7781 32.60933 0.0 32.60933 curveto 0.7781 32.60933 1.40889 33.24013 1.40889 34.01823 curveto closepath 0.0 34.01823 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 0.0 34.01823 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -28.05182 -17.00911 moveto -28.05182 -16.23099 -28.68262 -15.6002 -29.46072 -15.6002 curveto -30.23885 -15.6002 -30.86963 -16.23099 -30.86963 -17.00911 curveto -30.86963 -17.78722 -30.23885 -18.41801 -29.46072 -18.41801 curveto -28.68262 -18.41801 -28.05182 -17.78722 -28.05182 -17.00911 curveto closepath -29.46072 -17.00911 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 -29.46072 -17.00911 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore 0 TeXcolorgray restore save 0.79701 pgfw 1 0 0 TeXcolorrgb 34.37086 -19.84395 moveto 0.0 39.68793 lineto -34.37086 -19.84395 lineto closepath pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 35.77977 -19.84395 moveto 35.77977 -19.06584 35.14897 -18.43506 34.37086 -18.43506 curveto 33.59274 -18.43506 32.96196 -19.06584 32.96196 -19.84395 curveto 32.96196 -20.62207 33.59274 -21.25285 34.37086 -21.25285 curveto 35.14897 -21.25285 35.77977 -20.62207 35.77977 -19.84395 curveto closepath 34.37086 -19.84395 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 34.37086 -19.84395 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 1.40889 39.68793 moveto 1.40889 40.46605 0.7781 41.09683 0.0 41.09683 curveto -0.7781 41.09683 -1.40889 40.46605 -1.40889 39.68793 curveto -1.40889 38.90982 -0.7781 38.27904 0.0 38.27904 curveto 0.7781 38.27904 1.40889 38.90982 1.40889 39.68793 curveto closepath 0.0 39.68793 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 0.0 39.68793 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -32.96196 -19.84395 moveto -32.96196 -19.06584 -33.59274 -18.43506 -34.37086 -18.43506 curveto -35.14897 -18.43506 -35.77977 -19.06584 -35.77977 -19.84395 curveto -35.77977 -20.62207 -35.14897 -21.25285 -34.37086 -21.25285 curveto -33.59274 -21.25285 -32.96196 -20.62207 -32.96196 -19.84395 curveto closepath -34.37086 -19.84395 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 -34.37086 -19.84395 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore 0 TeXcolorgray restore 61.37656 35.43565 moveto 23.40205 57.35985 -23.40205 57.35985 -61.37656 35.43565 curveto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 62.78546 35.43565 moveto 62.78546 36.21376 62.15466 36.84456 61.37656 36.84456 curveto 60.59843 36.84456 59.96765 36.21376 59.96765 35.43565 curveto 59.96765 34.65753 60.59843 34.02675 61.37656 34.02675 curveto 62.15466 34.02675 62.78546 34.65753 62.78546 35.43565 curveto closepath 61.37656 35.43565 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 61.37656 35.43565 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 0 TeXcolorgray restore save 0.79701 pgfw 1 0 0 TeXcolorrgb 62.11485 36.70964 moveto 64.88373 40.93161 68.6111 42.80971 73.65186 42.5228 curveto 71.37537 38.00482 67.89996 35.72551 62.84903 35.438 curveto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 75.06076 42.5228 moveto 75.06076 43.30092 74.42998 43.9317 73.65186 43.9317 curveto 72.87375 43.9317 72.24295 43.30092 72.24295 42.5228 curveto 72.24295 41.74469 72.87375 41.11389 73.65186 41.11389 curveto 74.42998 41.11389 75.06076 41.74469 75.06076 42.5228 curveto closepath 73.65186 42.5228 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 73.65186 42.5228 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore 0 TeXcolorgray restore 0.0 -70.87134 moveto 37.97449 -48.94714 61.37656 -8.41403 61.37656 35.43625 curveto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 1.40889 -70.87134 moveto 1.40889 -70.09323 0.7781 -69.46243 0.0 -69.46243 curveto -0.7781 -69.46243 -1.40889 -70.09323 -1.40889 -70.87134 curveto -1.40889 -71.64946 -0.7781 -72.28024 0.0 -72.28024 curveto 0.7781 -72.28024 1.40889 -71.64946 1.40889 -70.87134 curveto closepath 0.0 -70.87134 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 0.0 -70.87134 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 0 TeXcolorgray restore save 0.79701 pgfw 1 0 0 TeXcolorrgb 0.73413 -72.14774 moveto 3.00623 -76.65683 2.76892 -80.82335 0.0 -85.04561 curveto -2.76892 -80.82335 -3.00623 -76.65683 -0.73413 -72.14774 curveto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 1.40889 -85.04561 moveto 1.40889 -84.26749 0.7781 -83.6367 0.0 -83.6367 curveto -0.7781 -83.6367 -1.40889 -84.26749 -1.40889 -85.04561 curveto -1.40889 -85.82372 -0.7781 -86.45451 0.0 -86.45451 curveto 0.7781 -86.45451 1.40889 -85.82372 1.40889 -85.04561 curveto closepath 0.0 -85.04561 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 0.0 -85.04561 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 1 0 0 TeXcolorrgb restore 0 TeXcolorgray restore -61.37656 35.43565 moveto -61.37656 -8.41463 -37.97449 -48.94774 0.0 -70.87193 curveto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -59.96765 35.43565 moveto -59.96765 36.21376 -60.59843 36.84456 -61.37656 36.84456 curveto -62.15466 36.84456 -62.78546 36.21376 -62.78546 35.43565 curveto -62.78546 34.65753 -62.15466 34.02675 -61.37656 34.02675 curveto -60.59843 34.02675 -59.96765 34.65753 -59.96765 35.43565 curveto closepath -61.37656 35.43565 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 -61.37656 35.43565 ] concat pgfs 1872 1530 a /pgfsc{}def/pgffc{}def 1872 1530 a 1872 1530 a 0 setgray 1872 1530 a 0 TeXcolorgray 0 TeXcolorgray 1872 1530 a pgfr restore 0 TeXcolorgray restore save 0.79701 pgfw 1 0 0 TeXcolorrgb -62.84903 35.438 moveto -67.89996 35.72551 -71.37537 38.00482 -73.65186 42.5228 curveto -68.6111 42.80971 -64.88373 40.93161 -62.11485 36.70964 curveto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -72.24295 42.5228 moveto -72.24295 43.30092 -72.87375 43.9317 -73.65186 43.9317 curveto -74.42998 43.9317 -75.06076 43.30092 -75.06076 42.5228 curveto -75.06076 41.74469 -74.42998 41.11389 -73.65186 41.11389 curveto -72.87375 41.11389 -72.24295 41.74469 -72.24295 42.5228 curveto closepath -73.65186 42.5228 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.3 0.0 0.0 0.3 -73.65186 42.5228 ] concat pgfs 1872 1530 a 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curveto 15.47026 50.66022 16.52155 51.71152 16.52155 53.00838 curveto closepath 14.17339 53.00838 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 14.17339 53.00838 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -6.36511 57.75693 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Fk(\000)p Ft(1)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -11.82523 53.00838 moveto -11.82523 54.30525 -12.87653 55.35654 -14.17339 55.35654 curveto -15.47026 55.35654 -16.52155 54.30525 -16.52155 53.00838 curveto -16.52155 51.71152 -15.47026 50.66022 -14.17339 50.66022 curveto -12.87653 50.66022 -11.82523 51.71152 -11.82523 53.00838 curveto closepath -14.17339 53.00838 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 -14.17339 53.00838 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -16.64586 31.48631 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Ft(1)1515 1011 y pgfr restore restore save save [1.0 0.0 0.0 1.0 -16.64854 10.05478 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Fk(\000)p Ft(2)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 0.0 moveto 2.34814 1.29686 1.29686 2.34814 0.0 2.34814 curveto -1.29686 2.34814 -2.34814 1.29686 -2.34814 0.0 curveto -2.34814 -1.29686 -1.29686 -2.34814 0.0 -2.34814 curveto 1.29686 -2.34814 2.34814 -1.29686 2.34814 0.0 curveto closepath 0.0 0.0 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 0.0 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -8.8997 -17.3836 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Ft(2)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 -28.3468 moveto 2.34814 -27.04994 1.29686 -25.99864 0.0 -25.99864 curveto -1.29686 -25.99864 -2.34814 -27.04994 -2.34814 -28.3468 curveto -2.34814 -29.64366 -1.29686 -30.69496 0.0 -30.69496 curveto 1.29686 -30.69496 2.34814 -29.64366 2.34814 -28.3468 curveto closepath 0.0 -28.3468 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 -28.3468 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -30.12837 -33.39177 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Fk(\000)p Ft(1)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -24.6115 -47.93387 moveto -24.6115 -46.63701 -25.66278 -45.58571 -26.95966 -45.58571 curveto -28.25652 -45.58571 -29.30782 -46.63701 -29.30782 -47.93387 curveto -29.30782 -49.23073 -28.25652 -50.28203 -26.95966 -50.28203 curveto -25.66278 -50.28203 -24.6115 -49.23073 -24.6115 -47.93387 curveto closepath -26.95966 -47.93387 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 -26.95966 -47.93387 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -30.71043 -66.99054 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Ft(1)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -14.31363 -79.62677 moveto -14.31363 -78.32991 -15.36491 -77.27861 -16.66179 -77.27861 curveto -17.95865 -77.27861 -19.00995 -78.32991 -19.00995 -79.62677 curveto -19.00995 -80.92363 -17.95865 -81.97493 -16.66179 -81.97493 curveto -15.36491 -81.97493 -14.31363 -80.92363 -14.31363 -79.62677 curveto closepath -16.66179 -79.62677 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 -16.66179 -79.62677 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -6.36511 -89.96555 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Fk(\000)p Ft(1)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 19.00995 -79.62677 moveto 19.00995 -78.32991 17.95865 -77.27861 16.66179 -77.27861 curveto 15.36491 -77.27861 14.31363 -78.32991 14.31363 -79.62677 curveto 14.31363 -80.92363 15.36491 -81.97493 16.66179 -81.97493 curveto 17.95865 -81.97493 19.00995 -80.92363 19.00995 -79.62677 curveto closepath 16.66179 -79.62677 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 16.66179 -79.62677 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 25.72903 -66.99054 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Ft(1)1515 1011 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 29.30782 -47.93387 moveto 29.30782 -46.63701 28.25652 -45.58571 26.95966 -45.58571 curveto 25.66278 -45.58571 24.6115 -46.63701 24.6115 -47.93387 curveto 24.6115 -49.23073 25.66278 -50.28203 26.95966 -50.28203 curveto 28.25652 -50.28203 29.30782 -49.23073 29.30782 -47.93387 curveto closepath 26.95966 -47.93387 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 26.95966 -47.93387 ] concat pgfs 1515 1011 a /pgfsc{}def/pgffc{}def 1515 1011 a 1515 1011 a 0 setgray 1515 1011 a 0 TeXcolorgray 0 TeXcolorgray 1515 1011 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 17.39815 -33.39178 ] concat pgfs 1515 1011 a 0 setgray 1515 1011 a Fk(\000)p Ft(1)1515 1011 y pgfr restore restore restore restore newpath restore pgfc 2171 1592 a 2171 1592 a 2171 1592 a pgfo save 0 setgray 0.3985 pgfw save save 1.19553 pgfw 0.0 0.0 moveto 0.0 28.3468 lineto 0.0 56.69362 lineto 0.0 85.04042 lineto 0.0 113.38724 lineto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def /pgfsc{}def/pgffc{}def 0 0 1 TeXcolorrgb save 0 TeXcolorgray /pgffc{0 setgray}def /pgfsc{}def/pgffc{}def 0 0 1 TeXcolorrgb 2.34814 0.0 moveto 2.34814 1.29686 1.29686 2.34814 0.0 2.34814 curveto -1.29686 2.34814 -2.34814 1.29686 -2.34814 0.0 curveto -2.34814 -1.29686 -1.29686 -2.34814 0.0 -2.34814 curveto 1.29686 -2.34814 2.34814 -1.29686 2.34814 0.0 curveto closepath 0.0 0.0 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray 0 0 1 TeXcolorrgb restore save [0.5 0.0 0.0 0.5 0.0 0.0 ] concat pgfs 2171 1592 a 0 0 1 setrgbcolor 2171 1592 a 0 0 1 TeXcolorrgb 0 0 1 TeXcolorrgb 2171 1592 a pgfr restore 0 TeXcolorgray 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -16.64854 11.37828 ] concat pgfs 2171 1592 a 0 setgray 2171 1592 a Fk(\000)p Ft(1)2171 1592 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 28.3468 moveto 2.34814 29.64366 1.29686 30.69496 0.0 30.69496 curveto -1.29686 30.69496 -2.34814 29.64366 -2.34814 28.3468 curveto -2.34814 27.04994 -1.29686 25.99864 0.0 25.99864 curveto 1.29686 25.99864 2.34814 27.04994 2.34814 28.3468 curveto closepath 0.0 28.3468 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 28.3468 ] concat pgfs 2171 1592 a /pgfsc{}def/pgffc{}def 2171 1592 a 2171 1592 a 0 setgray 2171 1592 a 0 TeXcolorgray 0 TeXcolorgray 2171 1592 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -8.8997 39.30998 ] concat pgfs 2171 1592 a 0 setgray 2171 1592 a Ft(1)2171 1592 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 56.69362 moveto 2.34814 57.99048 1.29686 59.04176 0.0 59.04176 curveto -1.29686 59.04176 -2.34814 57.99048 -2.34814 56.69362 curveto -2.34814 55.39674 -1.29686 54.34546 0.0 54.34546 curveto 1.29686 54.34546 2.34814 55.39674 2.34814 56.69362 curveto closepath 0.0 56.69362 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 56.69362 ] concat pgfs 2171 1592 a /pgfsc{}def/pgffc{}def 2171 1592 a 2171 1592 a 0 setgray 2171 1592 a 0 TeXcolorgray 0 TeXcolorgray 2171 1592 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -16.64854 68.0719 ] concat pgfs 2171 1592 a 0 setgray 2171 1592 a Fk(\000)p Ft(1)2171 1592 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 85.04042 moveto 2.34814 86.33728 1.29686 87.38858 0.0 87.38858 curveto -1.29686 87.38858 -2.34814 86.33728 -2.34814 85.04042 curveto -2.34814 83.74356 -1.29686 82.69226 0.0 82.69226 curveto 1.29686 82.69226 2.34814 83.74356 2.34814 85.04042 curveto closepath 0.0 85.04042 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 85.04042 ] concat pgfs 2171 1592 a /pgfsc{}def/pgffc{}def 2171 1592 a 2171 1592 a 0 setgray 2171 1592 a 0 TeXcolorgray 0 TeXcolorgray 2171 1592 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -8.8997 96.0036 ] concat pgfs 2171 1592 a 0 setgray 2171 1592 a Ft(1)2171 1592 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def /pgfsc{}def/pgffc{}def 0 0 1 TeXcolorrgb save 0 TeXcolorgray /pgffc{0 setgray}def /pgfsc{}def/pgffc{}def 0 0 1 TeXcolorrgb 2.34814 113.38724 moveto 2.34814 114.6841 1.29686 115.7354 0.0 115.7354 curveto -1.29686 115.7354 -2.34814 114.6841 -2.34814 113.38724 curveto -2.34814 112.09036 -1.29686 111.03908 0.0 111.03908 curveto 1.29686 111.03908 2.34814 112.09036 2.34814 113.38724 curveto closepath 0.0 113.38724 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray 0 0 1 TeXcolorrgb restore save [0.5 0.0 0.0 0.5 0.0 113.38724 ] concat pgfs 2171 1592 a 0 0 1 setrgbcolor 2171 1592 a 0 0 1 TeXcolorrgb 0 0 1 TeXcolorrgb 2171 1592 a pgfr restore 0 TeXcolorgray 0 TeXcolorgray restore restore restore newpath restore pgfc 2779 1177 a 2779 1177 a 2779 1177 a pgfo save 0 setgray 0.3985 pgfw save save 1.19553 pgfw 0.0 -28.3468 moveto 14.17339 -53.00838 lineto -14.17339 -53.00838 lineto -1.3189 -30.64175 lineto 0.0 -25.69977 moveto 0.0 0.0 lineto 0.0 28.3468 lineto 0.0 56.69362 lineto pgfstr save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 -28.3468 moveto 2.34814 -27.04994 1.29686 -25.99864 0.0 -25.99864 curveto -1.29686 -25.99864 -2.34814 -27.04994 -2.34814 -28.3468 curveto -2.34814 -29.64366 -1.29686 -30.69496 0.0 -30.69496 curveto 1.29686 -30.69496 2.34814 -29.64366 2.34814 -28.3468 curveto closepath 0.0 -28.3468 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 -28.3468 ] concat pgfs 2779 1177 a /pgfsc{}def/pgffc{}def 2779 1177 a 2779 1177 a 0 setgray 2779 1177 a 0 TeXcolorgray 0 TeXcolorgray 2779 1177 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 11.005 -36.75926 ] concat pgfs 2779 1177 a 0 setgray 2779 1177 a Ft(1)2779 1177 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 16.52155 -53.00838 moveto 16.52155 -51.71152 15.47026 -50.66022 14.17339 -50.66022 curveto 12.87653 -50.66022 11.82523 -51.71152 11.82523 -53.00838 curveto 11.82523 -54.30525 12.87653 -55.35654 14.17339 -55.35654 curveto 15.47026 -55.35654 16.52155 -54.30525 16.52155 -53.00838 curveto closepath 14.17339 -53.00838 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 14.17339 -53.00838 ] concat pgfs 2779 1177 a /pgfsc{}def/pgffc{}def 2779 1177 a 2779 1177 a 0 setgray 2779 1177 a 0 TeXcolorgray 0 TeXcolorgray 2779 1177 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -6.36511 -63.34715 ] concat pgfs 2779 1177 a 0 setgray 2779 1177 a Fk(\000)p Ft(1)2779 1177 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def -11.82523 -53.00838 moveto -11.82523 -51.71152 -12.87653 -50.66022 -14.17339 -50.66022 curveto -15.47026 -50.66022 -16.52155 -51.71152 -16.52155 -53.00838 curveto -16.52155 -54.30525 -15.47026 -55.35654 -14.17339 -55.35654 curveto -12.87653 -55.35654 -11.82523 -54.30525 -11.82523 -53.00838 curveto closepath -14.17339 -53.00838 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 -14.17339 -53.00838 ] concat pgfs 2779 1177 a /pgfsc{}def/pgffc{}def 2779 1177 a 2779 1177 a 0 setgray 2779 1177 a 0 TeXcolorgray 0 TeXcolorgray 2779 1177 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -16.64586 -37.90675 ] concat pgfs 2779 1177 a 0 setgray 2779 1177 a Ft(1)2779 1177 y pgfr restore restore save save [1.0 0.0 0.0 1.0 -16.64854 -15.645 ] concat pgfs 2779 1177 a 0 setgray 2779 1177 a Fk(\000)p Ft(2)2779 1177 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 0.0 moveto 2.34814 1.29686 1.29686 2.34814 0.0 2.34814 curveto -1.29686 2.34814 -2.34814 1.29686 -2.34814 0.0 curveto -2.34814 -1.29686 -1.29686 -2.34814 0.0 -2.34814 curveto 1.29686 -2.34814 2.34814 -1.29686 2.34814 0.0 curveto closepath 0.0 0.0 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 0.0 ] concat pgfs 2779 1177 a /pgfsc{}def/pgffc{}def 2779 1177 a 2779 1177 a 0 setgray 2779 1177 a 0 TeXcolorgray 0 TeXcolorgray 2779 1177 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -8.8997 10.96317 ] concat pgfs 2779 1177 a 0 setgray 2779 1177 a Ft(2)2779 1177 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def save /pgfsc{}def/pgffc{}def 0 TeXcolorgray /pgffc{0 setgray}def 2.34814 28.3468 moveto 2.34814 29.64366 1.29686 30.69496 0.0 30.69496 curveto -1.29686 30.69496 -2.34814 29.64366 -2.34814 28.3468 curveto -2.34814 27.04994 -1.29686 25.99864 0.0 25.99864 curveto 1.29686 25.99864 2.34814 27.04994 2.34814 28.3468 curveto closepath 0.0 28.3468 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save [0.5 0.0 0.0 0.5 0.0 28.3468 ] concat pgfs 2779 1177 a /pgfsc{}def/pgffc{}def 2779 1177 a 2779 1177 a 0 setgray 2779 1177 a 0 TeXcolorgray 0 TeXcolorgray 2779 1177 a pgfr restore 0 TeXcolorgray restore save save [1.0 0.0 0.0 1.0 -16.64854 39.72508 ] concat pgfs 2779 1177 a 0 setgray 2779 1177 a Fk(\000)p Ft(2)2779 1177 y pgfr restore restore save 0 TeXcolorgray /pgffc{0 setgray}def /pgfsc{}def/pgffc{}def 0 0 1 TeXcolorrgb save 0 TeXcolorgray /pgffc{0 setgray}def /pgfsc{}def/pgffc{}def 0 0 1 TeXcolorrgb 2.34814 56.69362 moveto 2.34814 57.99048 1.29686 59.04176 0.0 59.04176 curveto -1.29686 59.04176 -2.34814 57.99048 -2.34814 56.69362 curveto -2.34814 55.39674 -1.29686 54.34546 0.0 54.34546 curveto 1.29686 54.34546 2.34814 55.39674 2.34814 56.69362 curveto closepath 0.0 56.69362 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray 0 0 1 TeXcolorrgb restore save [0.5 0.0 0.0 0.5 0.0 56.69362 ] concat pgfs 2779 1177 a 0 0 1 setrgbcolor 2779 1177 a 0 0 1 TeXcolorrgb 0 0 1 TeXcolorrgb 2779 1177 a pgfr restore 0 TeXcolorgray 0 TeXcolorgray restore restore restore newpath restore pgfc 440 2027 a Ft(Figure)30 b(2:)815 2027 y SDict begin H.S end 815 2027 a 815 2027 a SDict begin H.R end 815 2027 a 815 2027 a SDict begin [/View [/XYZ H.V]/Dest (figure.2) cvn /DEST pdfmark end 815 2027 a Ft(Examples)e(of)j(eac)n(h)g(of)g(the)h(four)f(t)n(yp)r(es)g(of)g (construction)f(of)h Fr(t)h Ft(in)e(the)i(pro)r(of)440 2127 y(of)d(Lemma)834 2127 y SDict begin H.S end 834 2127 a 0 TeXcolorgray Ft(14)p 0 TeXcolorgray 917 2073 a SDict begin H.R end 917 2073 a 917 2127 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 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Fh(i)1554 4746 y Ft(\))23 b(=)g(3)p Fr(\013)p Ft(\()p Fr(L)p Ft(\()p Fk(D)1977 4758 y Fh(i)2005 4746 y Ft(\)\))c Fk(\000)f Fr(t)p Ft(\()p Fk(D)2297 4758 y Fh(i)2325 4746 y Ft(\))440 4907 y(and)27 b(write)g Fr(\020)6 b Ft(\()p Fr(Q)p Ft(\))24 b(=)1097 4844 y Fi(P)1185 4865 y Fh(`)1185 4932 y(i)p Fp(=1)1310 4907 y Fr(\020)6 b Ft(\()p Fk(D)1448 4919 y Fh(i)1477 4907 y Ft(\))28 b(so)f(that)1042 5175 y Fr(t)p Ft(\()p Fr(H)7 b Ft(\))23 b(=)1369 5072 y Fh(`)1323 5097 y Fi(X)1329 5273 y Fh(i)p Fp(=1)1457 5175 y Fr(t)p Ft(\()p Fk(D)1583 5187 y Fh(i)1611 5175 y Ft(\))g(=)1754 5033 y Fi( )1819 5175 y Ft(3)1921 5072 y Fh(`)1875 5097 y Fi(X)1881 5273 y Fh(i)p Fp(=1)2009 5175 y Fr(\013)p Ft(\()p Fr(L)p Ft(\()p Fk(D)2247 5187 y Fh(i)2275 5175 y Ft(\)\))2339 5033 y Fi(!)2424 5175 y Fk(\000)18 b Fr(\020)6 b Ft(\()p Fr(Q)p Ft(\))p Fr(:)p 0 TeXcolorgray 1831 5662 a Ft(12)p 0 TeXcolorgray eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop 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