坐标扩展度与分层$k$-均匀超图

中文摘要 AI 辅助

设$\Palt=(\C,\T)$为一个$k$-调色板。对于$0\le t\le k-1$，其第$t$个坐标扩展度定义为：在所有$t$个坐标的选择以及对这些坐标的所有颜色分配中，完成固定颜色为可容许$k$-元组的剩余$k-t$个坐标的分配比例的最小值。对于$k$-图$F$，我们定义$\pi_t^{\ext}(F)$为所有不被$F$容许的调色板上该度的上确界。我们证明\\[ \pi_t^{\ext}(F)=0 \quad\text{当且仅当}\quad F\text{是}t\text{-分层的}. \\]我们还将$t$-分层性与消失阶、最小分层性、最大分层性和分层性联系起来。这些结果恢复并推广了Reiher、Rödl和Schacht以及Lamaison的先前刻画，并回答了Lamaison关于$3$-图的一个问题。在$t=0$时，参数$\pi_0^{\ext}(F)$是$(k-2)$-均匀Turán密度$\pi_{k-2}(F)$。对于每个$k\ge3$和$r\ge2$，我们构造一个有限的$k$-图$F_{k,r}$，使得\\( \pi_{k-2}(F_{k,r})=2(r-1)/rk^k. \\)因此$2/k^k$是单个禁止$k$-图的累积点。我们还证明了不满足Lin、Wang和Zhou的条件$\Sp$的$k$-图的最小密度为$4/(3k^k)$。最后，对于每个大小为$m$的可容许匹配，我们构造一个$k$-图，它满足每个坐标对的$\Sp$，没有消失阶，且密度为$2^m/k^k$。这反驳了Lin、Wang和Zhou对每个$k\ge3$的一个猜想。

英文摘要

Let $\Palt=(\C,\T)$ be a $k$-palette. For $0\le t\le k-1$, its $t$th coordinate-extension degree is the minimum, over every choice of $t$ coordinates and every assignment of colors to them, of the proportion of assignments to the remaining $k-t$ coordinates that complete the fixed colors to an admissible $k$-tuple. For a $k$-graph $F$, we define $π_t^{\ext}(F)$ as the supremum of this degree over all palettes not admitted by $F$. We prove that \[ π_t^{\ext}(F)=0 \quad\text{if and only if}\quad F\text{ is }t\text{-layered}. \] We also relate $t$-layeredness to vanishing orders, min-layeredness, max-layeredness, and layeredness. These results recover and extend previous characterizations of Reiher, Rödl, and Schacht and of Lamaison, and answer a question of Lamaison for $3$-graphs. At $t=0$, the parameter $π_0^{\ext}(F)$ is the $(k-2)$-uniform Turán density $π_{k-2}(F)$. For every $k\ge3$ and $r\ge2$, we construct a finite $k$-graph $F_{k,r}$ with \( π_{k-2}(F_{k,r})=2(r-1)/rk^k. \) Thus $2/k^k$ is an accumulation point for single forbidden $k$-graphs. We also show that the least density of a $k$-graph that fails condition $\Sp$ of Lin, Wang and Zhou is $4/(3k^k)$. Finally, for every admissible matching of size $m$, we construct a $k$-graph that satisfies $\Sp$ for every coordinate pair, has no vanishing order, and has density $2^m/k^k$. This disproves a conjecture of Lin, Wang and Zhou for every $k\ge3$.

发表机构

• School of Mathematics, Nanjing University（南京大学数学系）

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