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This is a Quantitative Study study.

October 8, 2025Open Access

A semidefinite programming approach to cross 2-intersecting families

Key Points

The result establishes that the product of the sizes of the families is bounded above, resolving a major open problem.
Specifically, the inequality states that the product of sizes does not exceed the square of binomial coefficients.
The study explores both combinatorial configurations and their implications in product measure settings, broadening understanding.
A robust methodology utilizing semidefinite programming provides the foundation for these findings and proofs.

Abstract

Let k 2 and n 3 (k-1). Let F and G be families of k-element subsets of an n-element set. Suppose that |F G| 2 for all F and G. We show that |F||G|-2k-2², 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.

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