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- W2952022207 abstract "A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of $K_n$ is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large $p$, every maximum triangle-free subgraph of $G(n,p)$ is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for $p > K sqrt{log n/n}$ for some constant $K$, and apart from the value of the constant this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by $F_5$, which sometimes called as the generalized triangle, the 3-uniform hypergraph with vertex set {a,b,c,d,e} and edge set {abc, ade, bde}. One of the first extremal results in extremal hypergraph theory is by Frankl and F{u}redi, who proved that the maximum 3-uniform hypergraph on n vertices containing no copy of $F_5$ is tripartite for n>3000. A natural question is for what p is every maximum $F_5$-free subhypergraph of $G^3(n,p)$ w.h.p. tripartite. We show this holds for $p>Klog n/n$ for some constant K and does not hold for $p=0.1sqrt{log n}/n$." @default.
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- W2952022207 date "2013-10-05" @default.
- W2952022207 modified "2023-09-27" @default.
- W2952022207 title "Mantel's Theorem for Random Hypergraphs" @default.
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