埃尔德什匹配猜想的近最优线性范围
A Near-Optimal Linear Range for the Erdős Matching Conjecture
- Institute for Mathematical Sciences, Renmin University of China（中国人民大学数学科学研究所）
- Extremal Combinatorics and Probability Group, Institute for Basic Science（基础科学研究院极值组合与概率组）
- Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究确定了埃尔德什匹配猜想中第二种构造为极值的近最优线性范围，将相关线性系数从(5k-2)/3降至k+1，还证明了稳定性定理，其关键要素为概率刚性命题。
AI中文摘要：
埃尔德什匹配猜想由两种相互竞争的排除s+1条不相交边的方式决定：一种是将所有边集中在少于k(s+1)个顶点上，另一种是迫使每条边与一个固定的s元集合相交。我们确定了第二种构造为极值的近最优范围。对于每个固定的k≥2，存在s₀(k)，使得当s≥s₀(k)且n≥(k+1)s时，每个满足ν(𝒫)≤s的𝒫⊆binom([n],k)都有|𝒫|≤binom(n,k)-binom(n-s,k)，仅当𝒫是所有与固定s元集合相交的k元集合构成的族时等号成立。这将之前最佳的一般线性系数从(5k-2)/3降至k+1。由于两种猜想的构造在n=(ρₖ+o(1))s时交换渐近优势，其中ρₖ∈(k,k+1)，我们的范围位于不可逾越的障碍上方不到一个单位处。我们还证明了一个稳定性定理，表明在整个该范围内，覆盖族是唯一的近极值构型。我们证明中的一个关键要素是概率刚性命题，它迫使近极值分数覆盖几乎是整数值的。
英文摘要：
The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such that, whenever $s\ge s_0(k)$ and $n\ge(k+1)s$, every $\mathcal{F}\subseteq\binom{[n]}k$ with $ν(\mathcal{F})\le s$ satisfies \[ |\mathcal{F}|\le\binom nk-\binom{n-s}k, \] with equality only for the family of all $k$-sets meeting a fixed $s$-set. This improves the best previous general linear coefficient from $(5k-2)/3$ to $k+1$. In particular, the parameterized form of our argument further lowers the coefficient to $k+0.6$ for $k\ge5$. Since the two conjectured constructions exchange asymptotic dominance at $n=(ρ_k+o(1))s$ for a coefficient $ρ_k\in(k,k+1)$, our range lies less than one unit above the unavoidable barrier. We also prove a stability theorem showing that cover families are the only near-extremal configurations throughout this range. A key ingredient in our proof is a probabilistic rigidity statement which forces near-extremal fractional covers to be almost integral.
