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Stability results for graphs containing a critical edge

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The classical stability theorem of Erd\H{o}s and Simonovits states that, for any fixed graph $H$ with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n 2)$ of the maximal possible number of edges can be made into the $k$-partite Tur\’{a}n graph by adding and deleting $o(n 2)$ edges. We prove sharper quantitative results for graphs $H$ with a critical edge, showing how the $o(n 2)$ terms depend on each other. In many cases, these results are optimal to within a constant factor. We also discuss other recent results in a similar vein and some motivation for providing tighter bounds.

This talk is part of the Combinatorics Seminar series.

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