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- W4300535521 abstract "The study of intersecting structures is central to extremal combinatorics. A family of permutations $mathcal{F} subset S_n$ is emph{$t$-intersecting} if any two permutations in $mathcal{F}$ agree on some $t$ indices, and is emph{trivial} if all permutations in $mathcal{F}$ agree on the same $t$ indices. A $k$-uniform hypergraph is emph{$t$-intersecting} if any two of its edges have $t$ vertices in common, and emph{trivial} if all its edges share the same $t$ vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for $n$ sufficiently large with respect to $t$, the largest $t$-intersecting families in $S_n$ are the trivial ones. The classic ErdH{o}s--Ko--Rado theorem shows that the largest $t$-intersecting $k$-uniform hypergraphs are also trivial when $n$ is large. We determine the emph{typical} structure of $t$-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollob'as set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces." @default.
- W4300535521 created "2022-10-03" @default.
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- W4300535521 date "2014-08-11" @default.
- W4300535521 modified "2023-09-24" @default.
- W4300535521 title "Intersecting families of discrete structures are typically trivial" @default.
- W4300535521 doi "https://doi.org/10.48550/arxiv.1408.2559" @default.
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