Coordinate-extension degrees and layered $k$-uniform hypergraphs

Abstract

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 $k\ge3$.