On a weaker notion of cross t-intersecting families

2026/01/28 by Jiangdong Ai, Ming Chen, Seokbeom Kim +1 · 1 voice
Mathematics · #math.CO

paper · pdf

Abstract

We prove that if two families F ⊆ \binom[n]k and F' ⊆ \binom[n]k' satisfy ∑_{1 ≤ i, j ≤ ℓ} | F_i ∩ F_j' | ≥ ℓ²t - ℓ +1 for every choice of distinct F₁, …, F_ℓ ∈ F and F₁', …, F_ℓ' ∈ F', then | F | ⋅ | F' | ≤ \binomn-tk-t \binomn-tk'-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.

Citations

• A new intersection condition in extremal set theory
• INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS

Discussions

• #New_arXiv_paper Jiangdong Ai, Ming Chen, *Seokbeom Kim*, and Hyunwoo Lee, On a weaker notion of cross t-intersecting families, 2025. arxiv.org/abs/2601.20516 [bsky, 0 points, 0 comments]

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