Non-trivial intersecting uniform sub-families of hereditary families

For a family F of sets, let μ(F ) denote the size of a smallest set in F that is not a subset of any other set in F , and for any positive integer r, let F (r) denote the family of r-element sets in F . We say that a family A is of Hilton–Milner (HM) type if for some A ∈ A, all sets in A \ {A} have a common element x ̸∈ A and intersect A. We show that if a hereditary family H is compressed and μ(H) ≥ 2r ≥ 4, then the HM-type family {A ∈ H(r): 1 ∈ A, A∩[2,r+1] ̸= ∅}∪{[2,r+1]}is a largest non-trivial intersecting sub-family of H(r); this generalises a well-known result of Hilton and Milner. We demonstrate that for any r ≥ 3 and m ≥ 2r, there exist non-compressed hereditary families H with μ(H) = m such that no largest non-trivial intersecting sub-family of H(r) is of HM type, and we suggest two conjectures about the extremal structures for arbitrary hereditary families.peer-reviewe

Non-trivial 3-wise intersecting uniform families

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every $\epsilon>0$ there exists $n_0$ such that if $n>n_0$ and $\frac25+\epsilon<\frac kn<\frac 12-\epsilon$ then the maximum size is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.Comment: 12 page

Counting Intersecting and Pairs of Cross-Intersecting Families

A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\H{o}s, Ko, and Rado determines the maximum size of an intersecting family of $k$-subsets of $\{1,\ldots, n\}$. In this paper we study the following problem: how many intersecting families of $k$-subsets of $\{1,\ldots, n\}$ are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we determine this quantity asymptotically for $n\ge 2k+2+2\sqrt{k\log k}$ and $k\to \infty$. Moreover, under the same assumptions we also determine asymptotically the number of {\it non-trivial} intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families

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