← 研踪 PaperTrailarXiv:2610.35136 · v1
Sharp Bounds for Graph-Codes Avoiding Complete Bipartite Differences
首次提交 2026-10-26 18:08
版本更新 2026-10-26 18:08
雷达观测 2026-10-29 00:36
摘要
Let $d_{K_{s,t}}(n)$ be the largest fraction of all graphs on $[n]$ in a family containing no two distinct members whose symmetric difference is a copy of $K_{s,t}$. We determine this maximum density for every fixed pair of positive integers $s,t$. If both part sizes are odd, it is known that the density is exactly $1/2$. Otherwise, let $2h$ be the largest even part size. We prove $$ d_{K_{s,t}}(n)=Θ_{s,t}(n^{-h}). $$ For $h>1$, this improves the general $O_{s,t}(1/n)$ upper bound obtained from Alon's two-copy gluing theorem~\cite{Alon2024}. The sharp bounds for even stars were already proved by Alon and are recovered as special cases. The upper bound uses the Frankl--Füredi forbidden-intersection theorem. For the lower bound, we construct binary linear codes using BCH codes. When exactly one part size is even, we assign to each edge the sum of the binary vectors assigned to its endpoints. When both part sizes are even, we identify these vectors with elements of a finite field and assign to each edge the product of its endpoint elements. In both cases, every copy of $K_{s,t}$ has a nonzero syndrome.
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-10-26 18:08 · 最新观测版本
Sharp Bounds for Graph-Codes Avoiding Complete Bipartite Differences
Let $d_{K_{s,t}}(n)$ be the largest fraction of all graphs on $[n]$ in a family containing no two distinct members whose symmetric difference is a copy of $K_{s,t}$. We determine this maximum density for every fixed pair of positive integers $s,t$. If both part sizes are odd, it is known that the density is exactly $1/2$. Otherwise, let $2h$ be the largest even part size. We prove $$ d_{K_{s,t}}(n)=Θ_{s,t}(n^{-h}). $$ For $h>1$, this improves the general $O_{s,t}(1/n)$ upper bound obtained from Alon's two-copy gluing theorem~\cite{Alon2024}. The sharp bounds for even stars were already proved by Alon and are recovered as special cases. The upper bound uses the Frankl--Füredi forbidden-intersection theorem. For the lower bound, we construct binary linear codes using BCH codes. When exactly one part size is even, we assign to each edge the sum of the binary vectors assigned to its endpoints. When both part sizes are even, we identify these vectors with elements of a finite field and assign to each edge the product of its endpoint elements. In both cases, every copy of $K_{s,t}$ has a nonzero syndrome.
查看此版本解读 →查看此版本原文 ↗