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- W1520390178 abstract "We define the notion of stability for a monotone property of set systems. This phenomenon encompasses some classical results in combinatorics, foremost among them the ErdÅs-Simonovits stability theorem. A triangle is a family of three sets $A, B, C$ such that $A cap B$, $B cap C$, $C cap A$ are each nonempty, and $A cap B cap C= emptyset$. We prove the following new theorem about the stability of triangle-free set systems. Fix $rge 3$. For every $delta >0$, there exist $epsilon >0$ and $n_0=n_0(epsilon , r)$ such that the following holds for all $n>n_0$: if $|X|=n$ and $mathcal {G}$ is a triangle-free family of $r$-sets of $X$ containing at least $(1-epsilon ){n-1 choose r-1}$ members, then there exists an $(n-1)$-set $S subset X$ which contains fewer than $delta {n-1 choose r-1}$ members of $mathcal {G}$. This is one of the first stability theorems for a nontrivial problem in extremal set theory. Indeed, the corresponding extremal result, that for $n ge 3r/2>4$ every triangle-free family $mathcal {G}$ of $r$-sets of $X$ has size at most ${n-1 choose r-1}$, was a longstanding conjecture of ErdÅs (open since 1971) that was only recently settled by Mubayi and Verstraëte (2005) for all $nge 3r/2$." @default.
- W1520390178 created "2016-06-24" @default.
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- W1520390178 date "2006-08-16" @default.
- W1520390178 modified "2023-10-14" @default.
- W1520390178 title "Structure and stability of triangle-free set systems" @default.
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- W1520390178 doi "https://doi.org/10.1090/s0002-9947-06-04009-8" @default.
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