2024/04/19 by Ryan R. Martin, Balázs Patkós, Martin, Ryan R. +1
Mathematics · #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.12971
openalex publication_date 2024/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Erd\H os Matching Conjecture states that the maximum size f(n,k,s) of a family F⊆ \binom[n]k that does not contain s pairwise disjoint sets is max\|Ak,s|,|Bn,k,s|\, where Ak,s=\binom[sk-1]k and Bn,k,s=\B∈ \binom[n]k:B∩ [s-1]≠ ∅\. The case s=2 is simply the Erdős-Ko-Rado theorem on intersecting families and is well understood. The case n=sk was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if k,s are fixed and n is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for n∈ [sk,sk+cs,k] if s is large enough compared to k. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most s in a similar range of parameters. In this short note, we are concerned with the case s≥ 3 fixed, k tending to infinity and n∈\sk,sk+1\. For n=sk, we show the stability of the unique extremal construction of size \binomsk-1k=(s-1)/(s)\binomskk with respect to minimal degree. As a consequence we derive limk→ ∞\fracf(sk+1,k,s)\binomsk+1k<(s-1)/(s)-εs for some positive constant εs which depends only on s.