Bounded fractional intersecting families are linear in size

2024/02/22 by Balachandran, Niranjan, Das, Shagnik, Sankarnarayanan, Brahadeesh
#05D05 (Primary) 03E05 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.14981

Abstract

Using the sunflower method, we show that if θ∈ (0,1) ∩ ℚ and F is a O(n^1/3)-bounded θ-intersecting family over [n], then | F | = O(n), and that if F is o(n^1/3)-bounded, then | F | ≤ ((3)/(2) + o(1))n. This partially solves a conjecture of Balachandran, Mathew and Mishra that any θ-intersecting family over [n] has size at most linear in n, in the regime where we have no very large sets.

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