Intersecting Families of Permutations

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.Comment: 'Erratum' section added. Yuval Filmus has recently pointed out that the 'Generalised Birkhoff theorem', Theorem 29, is false for k > 1, and so is Theorem 27 for k > 1. An alternative proof of the equality part of the Deza-Frankl conjecture is referenced, bypassing the need for Theorems 27 and 2

Intersecting families of discrete structures are typically trivial

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 Erd\H{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.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira result. Update 2: corrected statement of the unpublished Hamm--Kahn result, and slightly modified notation in Theorem 1.6 Update 3: new title, updated citations, and some minor correction

Intersecting families of permutations

AbstractLet Sn be the symmetric group on the set X={1,2,…,n}. A subset S of Sn is intersecting if for any two permutations g and h in S, g(x)=h(x) for some x∈X (that is g and h agree on x). Deza and Frankl (J. Combin. Theory Ser. A 22 (1977) 352) proved that if S⊆Sn is intersecting then |S|≤(n−1)!. This bound is met by taking S to be a coset of a stabiliser of a point. We show that these are the only largest intersecting sets of permutations

Fourier analysis on finite groups and the Lov\'asz theta-number of Cayley graphs

We apply Fourier analysis on finite groups to obtain simplified formulations for the Lov\'asz theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a recent article proving a version of the Erd\H{o}s-Ko-Rado theorem for $k$-intersecting families of permutations. We also introduce a $q$-analog of the notion of $k$-intersecting families of permutations, and we verify a few cases of the corresponding Erd\H{o}s-Ko-Rado assertion by computer.Comment: 9 pages, 0 figure

Cross-intersecting families of permutations

For positive integers r and n with r n, let Pr,n be the family of all sets {(1, y1), (2, y2),. . . , (r, yr)} such that y1, y2,..., yr are distinct elements of [n]={1, 2,...,n}. Pn,n describes permutations of [n]. For r < n, Pr,n describes permutations of r-element subsets of [n]. Families A1,A2,...,Ak of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in Ai intersects any set in Aj. For any r, n and k 2, we determine the cases in which the sum of sizes of cross-intersecting sub-families A1,A2,...,Ak of Pr,n is a maximum, hence solving a recent conjecture.peer-reviewe