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On a weaker notion of cross $t$-intersecting families

Jiangdong Ai, Ming Chen, Seokbeom Kim, Hyunwoo Lee

TL;DR

The paper studies weakening cross $t$-intersecting families to the $\ell$-weakly cross $t$-intersecting setting and proves that for sufficiently large $n$, if $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ are $\ell$-weakly cross $t$-intersecting, then $|\mathcal{F}|\,|\mathcal{F}'| \le \binom{n-t}{k-t}\binom{n-t}{k'-t}$. The proof combines sunflower techniques with kernels of size $t$ and Erdős' Matching Theorem to manage cases where large sunflowers are absent, leading to a contradiction if the product exceeds the bound. Equality configurations arise from fixing a $t$-subset and taking all sets containing it, demonstrating tightness. Overall, the result extends Pyber's theorem on cross $1$-intersecting families and highlights the robustness of product bounds under weaker intersection conditions in extremal combinatorics.

Abstract

We prove that if two families $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ satisfy $\sum_{1 \leq i, j \leq \ell} \lvert F_i \cap F_j' \rvert \geq \ell^2t - \ell +1$ for every choice of distinct $F_1, \ldots, F_\ell \in \mathcal{F}$ and $F_1', \ldots, F_\ell' \in \mathcal{F}'$, then $\lvert \mathcal{F} \rvert \cdot \lvert \mathcal{F}' \rvert \leq \binom{n-t}{k-t} \binom{n-t}{k'-t}$, provided that $n$ is sufficiently large. This extends a celebrated theorem of Pyber for large $n$, which determines the tight upper bound for the product of the sizes of cross $1$-intersecting families.

On a weaker notion of cross $t$-intersecting families

TL;DR

The paper studies weakening cross -intersecting families to the -weakly cross -intersecting setting and proves that for sufficiently large , if and are -weakly cross -intersecting, then . The proof combines sunflower techniques with kernels of size and Erdős' Matching Theorem to manage cases where large sunflowers are absent, leading to a contradiction if the product exceeds the bound. Equality configurations arise from fixing a -subset and taking all sets containing it, demonstrating tightness. Overall, the result extends Pyber's theorem on cross -intersecting families and highlights the robustness of product bounds under weaker intersection conditions in extremal combinatorics.

Abstract

We prove that if two families and satisfy for every choice of distinct and , then , provided that is sufficiently large. This extends a celebrated theorem of Pyber for large , which determines the tight upper bound for the product of the sizes of cross -intersecting families.
Paper Structure (2 sections, 3 theorems, 12 equations)

This paper contains 2 sections, 3 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Proof of \ref{['weakly-t-cross']}

Key Result

Theorem 1.1

Let $n$ and $k$ be positive integers. Suppose that $\mathcal{F} \subseteq \binom{[n]}{k}$ satisfies for every choice of $\ell$ distinct members $F_1, \ldots, F_\ell \in \mathcal{F}$. Then $\lvert \mathcal{F} \rvert \leq \binom{n-1}{k-1}$ provided that $n$ is sufficiently large. Furthermore, the bound $\binom{\ell-1}{2}+1$ in the assumption is best possible.

Theorems & Definitions (6)

  • Theorem 1.1: FKN
  • Theorem 1.2
  • Claim 2.1
  • Claim 2.2
  • Theorem 2.3: Erdős' Matching Theorem, MR260599
  • Claim 2.4