Regular Intersecting Families

TL;DR

This paper studies intersecting families of k-element subsets of an n-set, focusing on their size and element distribution, revealing that such families are small when k grows much slower than n.

Contribution

It characterizes the size and distribution properties of regular intersecting families, especially their asymptotic behavior as n and k vary.

Findings

Intersects families are small when k=o(n).

Distribution of elements in the family is approximately uniform.

Provides bounds on the size of regular intersecting families.

Abstract

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the same (or approximately the same) number of members of $\mathcal{F}$. In particular, we show that we can guarantee $|\mathcal{F}| = o({n-1\choose k-1})$ if and only if $k=o(n)$.