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This is a Quantitative Study study.

October 8, 2025Open Access

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

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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.

Implication

This analysis reveals bounds on the sizes of families of k-element subsets, suggesting critical configurations in combinatorial settings.

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|≤n-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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Cite This Study

Tanaka et al. (2025) studied this question.

synapsesocial.com/papers/68e62de1a8c0c6d45873ff07 https://doi.org/10.48550/arxiv.2503.14844

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Implication

This analysis reveals bounds on the sizes of families of k-element subsets, suggesting critical configurations in combinatorial settings.

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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