2024/11/06 by Yunrong Wu, Wu, Yongjiang, YB Li +7
Mathematics · #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2411.03674
The well-known Erdős--Ko--Rado theorem states that for n> 2k, every intersecting family of k-sets of [n]:=\1,… ,n\ has at most n-1 \choose k-1 sets, and the extremal family consists of all k-sets containing a fixed element (called a full star). The Hilton--Milner theorem provides a stability result by determining the maximum size of a uniform intersecting family that is not a subfamily of a full star. The further stabilities were studied by Han and Kohayakawa (2017) and Huang and Peng (2024). Two families F and G are called cross-intersecting if for every F∈ F and G∈ G, the intersection F∩ G is non-empty. Let k ≥ 1, t≥ 0 and n ≥ 2 k+t be integers. Frankl (2016) proved that if F ⊆\binom[n]k+t and G ⊆\binom[n]k are cross-intersecting families, and F is non-empty and (t+1)-intersecting, then |F|+|G| ≤\binomnk-\binomn-k-tk+1. Recently, Wu (2023) sharpened Frankl's result by establishing a stability variant. The aim of this paper is two-fold. Inspired by the above results, we first prove a further stability variant that generalizes both Frankl's result and Wu's result. Secondly, as an interesting application, we illustrate that the aforementioned results on cross-intersecting families could be used to establish the stability results of the Erdős--Ko--Rado theorem. More precisely, we present new short proofs of the Hilton--Milner theorem, the Han--Kohayakawa theorem and the Huang--Peng theorem. Our arguments are more straightforward, and it may be of independent interest.