Nearly Erd\H{o}s-Ko-Rado theorems

TL;DR

This paper extends the Erd ext{o}s-Ko-Rado theorem by determining the maximum size of nearly intersecting families of sets under a weakened intersection condition, especially for large n and in the context of t-intersecting families.

Contribution

It establishes the maximum size of families satisfying a relaxed intersection condition, generalizing previous results to larger families and the t-intersecting case for sufficiently large n.

Findings

Maximum size of nearly intersecting families determined for large n.

Generalization to t-intersecting families achieved.

Weaker intersection condition still bounds family size similarly.

Abstract

If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\leq n$ then $|\mathcal{F}|\leq {n-1\choose k-1}$ holds by the celebrated Erd\H{o}s-Ko-Rado theorem. But an intersecting family obviously satisfies the condition $${\ell \choose 2}\leq \sum_{1\leq i<j\leq \ell}|F_i\cap F_j| $$ for any $\ell$ distinct members of the family. It has been proved in [5] that even if ${\ell \choose 2}$ is replaced by ${\ell -1 \choose 2}+1$ the conclusion $|\mathcal{F}|\leq {n-1\choose k-1}$ remains valid for large $n$. However the 1 cannot be omitted, because there is a larger family satisfying that weaker condition. In the present paper we determine the largest size of the family under this weaker condition when $n$ is sufficiently large. All of these are treated in the more general setting of $t$-intersecting families.