Kalai 关于紧树的猜想

中文摘要 AI 辅助

设 $r \ge 2$ 且 $t \ge 1$。本文证明：若 $T$ 是具有 $t$ 条边的 $r$-一致紧树，且 $H$ 是不含 $T$ 的 $r$-一致超图，则 $|E(H)|\le (t-1)|\sh H|/r$，其中 $\sh H$ 是 $H$ 的 $(r-1)$-影子。该界在无穷多情形下是紧的，并确立了 Kalai 猜想，其 $r=2$ 情形即为 Erdős-Sós 猜想。该证明由 GPT-6 Astra 发现，将其证明 Erdős-Sós 猜想的方法推广到了超图情形。值得注意的是，此前对 Erdős-Sós 猜想特殊情形的证明并不能推广到超图情形以给出紧的界。作者提出的关于平均度 $d \ge t-1\ge 0$ 的图中树副本数量的紧下界的 Erdős-Sós 猜想的加强版本仍然开放。

英文摘要

Let $r \ge 2$ and $t \ge 1$. It is shown that if $T$ is an $r$-uniform tight tree with $t$ edges and $H$ is a $T$-free $r$-uniform hypergraph, then $|E(H)|\le (t-1)|\sh H|/r$, where $\sh H$ is the $(r-1)$-shadow of $H$. \iffalse Equality holds only for $(n,t + r - 2,r)$-designs.\fi The bound is tight infinitely often, and establishes Kalai's Conjecture, whose $r=2$ case is the Erd\H os-Sós Conjecture. The proof was found by GPT-6 Astra, extending its method of proof for the Erd\H os-Sós conjecture to the hypergraph setting. It is noteworthy that previous proofs of special cases of the Erd\H os-Sós conjecture do not extend to give tights bounds in the hypergraph setting. A strengthening of the Erd\H os-Sós conjecture due to the authors about tight lower bounds on the number of copies of a tree in a graph with average degree $d \ge t-1\ge 0$ remains open.

发表机构

• University of Illinois, Chicago（伊利诺伊大学芝加哥分校）
• University of California, San Diego（加州大学圣地亚哥分校）

机构由 AI 辅助整理，请以论文原文为准。