A new intersection condition in extremal set theory

TL;DR

This work extends Erdős–Ko–Rado-type extremal theory to $(3,2,\\ell)$-intersecting families, examining both uniform and non-uniform settings for $\\ell \ge 2$. The authors develop shifting techniques to reduce to shifted configurations, establish a fundamental recurrence for the uniform case, and obtain tight upper bounds for $\\ell=2$ and $3$ with matching constructive lower bounds in the large-$n$ regime. In the non-uniform setting, they introduce a complementary intersection framework and up-shift methods to derive explicit bounds and extremal families, including detailed case analysis based on $x$ modulo 6. Collectively, the results advance understanding of how triple-intersection sums constrain family sizes and provide new tools for extremal set theory in generalized intersection problems.

Abstract

We call a family $\mathcal{F}$ $(3,2,\ell)$-intersecting if $|A \cap B|+|B \cap C|+|C \cap A| \geq \ell$ for all $A$, $B$, $C \in \mathcal{F}$. We try to look for the maximum size of such a family $\mathcal{F}$ in case when $\mathcal{F} \subset {[n] \choose k}$ or $\mathcal{F} \subset 2^{[n]}$. In the uniform case we show that if $\mathcal{F}$ is $(3,2,2)$-intersecting, then $\vert \mathcal{F} \vert \leq {n+1 \choose k-1}+{n \choose k-2}$ and if $\mathcal{F}$ is $(3,2,3)$-intersecting, then $|\mathcal{F}| \leq {n \choose k-1} + 2 {n \choose k-3} + 3 {n-1 \choose k-3}$. For the lower bound we construct a $(3,2,\ell)$-intersecting family and we show that this bound is sharp when $\ell=2$ or $3$ and $n$ is sufficiently large compared to $k$. In the non-uniform case we give an upper bound for a $(3,2,n-x)$-intersecting family, when $n$ is sufficiently large compared to $x$.

Theorems & Definitions (70)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.6
• Theorem 1.7
• Conjecture 1.8
• Theorem 1.9