小早川猜想与路和圈补图的团覆盖
Kohayakawa's conjecture and clique coverings of complements of paths and cycles
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中文总结 AI 辅助
该研究证明了小早川关于$w(s)$上确界的猜想，解决了德卡恩等人的团覆盖猜想，还独立证明了路和圈补图团覆盖的阶估计，运用了克内泽尔图性质与Lovász局部引理。
中文摘要 AI 辅助
对于$s\ge1$，设$G_s$是$[2s]$的$s$元子集与$(s-1)$元子集之间的二分图，邻接关系为不相交，令$w(s)$为$G_s$中诱导路上的$s$元子集的最大数量。我们证明对所有$s\geq6$，有$w(s)\ge \frac{4^s}{2048s^{5/2}}$。这意味着$\sup_{s\ge1}w(s)^{1/s}=4$，正如小早川（Kohayakawa）于1991年猜想的那样。随后，他的递归构造在克内泽尔图（Kneser graph）$KG(2r+1,r)$中给出了阶为$\Omega(4^r/r^{5/2})$的诱导路，并得到$\max\{\cc(\overline{P_n}),\\ \cc(\overline{C_n})\} \le \log_2 n+\frac52\log_2\log_2 n+O(1)$。结合已知的下界，这解决了德卡恩（de Caen）、格雷戈里（Gregory）和普尔曼（Pullman）于1985年提出的猜想，得出$\cc(\overline{P_n})=\log_2 n+\Theta(\log_2\log_2 n)$，$\cc(\overline{C_n})=\log_2 n+\Theta(\log_2\log_2 n)$。我们还给出了后一阶估计的独立证明，该证明使用了克内泽尔图的哈密顿性结果和由Lovász局部引理证明的关键引理。
英文摘要
For $s\ge1$, let $G_s$ be the bipartite graph between the $s$-subsets and the $(s-1)$-subsets of $[2s]$, where adjacency means disjointness, and let $w(s)$ be the maximum number of $s$-subsets on an induced path in $G_s$. We prove $w(s)\ge \frac{4^s}{2048s^{5/2}}$ for all $s\geq 6$. This implies $\sup_{s\ge1}w(s)^{1/s}=4$, as conjectured by Kohayakawa (1991). His recursive construction then gives induced paths of order $Ω(4^r/r^{5/2})$ in the Kneser graph $KG(2r+1,r)$ and yields \[ \max\{\cc(\overline{P_n}),\ \cc(\overline{C_n})\} \le \log_2 n+\frac52\log_2\log_2 n+O(1). \] Together with the known lower bounds, this settles a conjecture of de Caen, Gregory, and Pullman (1985) and gives \[ \cc(\overline{P_n})=\log_2 n+Θ(\log_2\log_2 n), \qquad \cc(\overline{C_n})=\log_2 n+Θ(\log_2\log_2 n). \] We also give an independent proof of the latter order estimates. It uses a Hamiltonicity result of Kneser graphs and a key lemma proved by the Lovász local lemma.
