Document Zbl 1348.05100

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Comb. Theory, Ser. B</a> <a href="/?q=in%3A359680" title="Articles in this Issue">121, 146-172 (2016)</a>. </div> <div class="abstract">Summary: For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $\mathrm{ex}(n, T, H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T = K_2$ is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">(i)</dt><dd class="zbmathjax-item1"> $\mathrm{ex}(n, K_3, C_5) \leq(1 + o(1)) \frac{\sqrt{3}}{2} n^{3 / 2}$,</dd><dt class="zbmathjax-item1">(ii)</dt><dd class="zbmathjax-item1"> for any fixed $m$, $s \geq 2 m - 2$ and $t \geq(s - 1)! + 1$, $\mathit{ex}(n, K_m, K_{s, t}) = \operatorname{\Theta}(n^{m -\binom{m}{2}/ s})$, and</dd><dt class="zbmathjax-item1">(iii)</dt><dd class="zbmathjax-item1"> for any two trees $H$ and $T$, $\mathrm{ex}(n, T, H) = \operatorname{\Theta}(n^m)$ where $m = m(T, H)$ is an integer depending on $H$ and $T$ (its precise definition is given in Section 1). </dd></dl> The first result improves (slightly) an estimate of <span class="zbmathjax-textit">B. Bollobás</span> and <span class="zbmathjax-textit">E. Győri</span> [Discrete Math. 308, No. 19, 4332–4336 (2008; <a href="/1152.05034">Zbl 1152.05034</a>)]. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6637919">Cited in <strong>23</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6637919">Cited in <strong>152</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C35" title="MSC2020">05C35</a> </td> <td class="space"> Extremal problems in graph theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aextremal+graph+theory">extremal graph theory</a>; <a href="/?q=ut%3A%5C%28H%5C%29-free+graphs">$H$-free graphs</a>; <a href="/?q=ut%3Aprojective+norm+graphs">projective norm graphs</a>; <a href="/?q=ut%3Acomplete+graphs">complete graphs</a>; <a href="/?q=ut%3Acomplete+bipartite+graphs">complete bipartite graphs</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1152.05034">Zbl 1152.05034</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1348.05100" data-ciurl="/ci/06637919" data-biburl="/bibtex/06637919.bib" data-amsurl="/amsrefs/06637919.bib" data-xmlurl="/xml/06637919.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06637919.pdf" title="Zbl 1348.05100 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jctb.2016.03.004" aria-label="DOI for “Many $T$ copies in $H$-free graphs”" title="10.1016/j.jctb.2016.03.004">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alon, N., On the number of subgraphs of prescribed type of graphs with a given number of edges, Israel J. 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