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A semidefinite programming approach to cross $2$-intersecting families

Hajime Tanaka, Norihide Tokushige

Abstract

Let $k\geq 2$ and $n\geq 3(k-1)$. Let $\mathcal{F}$ and $\mathcal{G}$ be families of $k$-element subsets of an $n$-element set. Suppose that $|F\cap G|\geq 2$ for all $F\in\mathcal{F}$ and $G\in\mathcal{G}$. We show that $|\mathcal{F}||\mathcal{G}|\leq\binom{n-2}{k-2}^2$, and determine the extremal configurations. This settles the last unsolved case of a recent result by Zhang and Wu (J. Combin. Theory Ser. B, 2025). We also obtain the corresponding result in the product measure setting. Our proof is done by solving semidefinite programming problems.

A semidefinite programming approach to cross $2$-intersecting families

Abstract

Let and . Let and be families of -element subsets of an -element set. Suppose that for all and . We show that , and determine the extremal configurations. This settles the last unsolved case of a recent result by Zhang and Wu (J. Combin. Theory Ser. B, 2025). We also obtain the corresponding result in the product measure setting. Our proof is done by solving semidefinite programming problems.

Paper Structure

This paper contains 9 sections, 6 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. The $k$-uniform setting
  3. Corresponding SDP problem
  4. Preliminaries
  5. Proof Theorem \ref{['thm:c2k']}
  6. The measure setting
  7. Corresponding SDP problem
  8. Preliminaries
  9. Proof Theorem \ref{['thm:c2p']}

Key Result

Theorem 1

Let $n\geq k\geq t\geq 1$ be integers, and let $n\geq (t+1)(k-t+1)$. Suppose that $\mathcal{F}\subset\binom{[n]}k$ is a $t$-intersecting family. Then $|\mathcal{F}|\leq\binom{n-t}{k-t}$. Moreover, if $n>(t+1)(k-t+1)$ and $|\mathcal{F}|=\binom{n-t}{k-t}$, then there exists a $t$-element subset $T\sub

Theorems & Definitions (8)

  • Theorem 1: EKRFcktW
  • Conjecture 1: T2013
  • Theorem 2
  • Theorem 3: AK1998DSFriedgut2008T2005
  • Conjecture 2
  • Theorem 4
  • Proposition 1
  • Proposition 2