Nearly Erdős-Ko-Rado theorems

TL;DR

The paper investigates the maximal size of $k$-uniform, $t$-intersecting families under a weakened intersection-sum condition, refining the classical Erdős–Ko–Rado framework. It provides a concrete extremal construction and proves its optimality for large $n$ using a sunflower-based upper-bound strategy combined with cross-intersection analysis, yielding an explicit bound on the size. The results extend the traditional EKR and FKN theorems to weaker constraints and connect to Erdős–Matching techniques, with additional open questions addressing tighter bounds, minimal $n$, and non-uniform generalizations.

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 Erdős-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.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 1
• Lemma 2
• Lemma 3