Ramsey goodness of $k$-uniform paths, or the lack thereof

Abstract

Given a pair of $k$-uniform hypergraphs $(G,H)$, the Ramsey number of $(G,H)$, denoted by $R(G,H)$, is the smallest integer $n$ such that in every red/blue-colouring of the edges of $K_n^{(k)}$ there exists a red copy of $G$ or a blue copy of $H$. Burr showed that, for any pair of graphs $(G,H)$, where $G$ is large and connected, $R(G,H) \geq (v(G)-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ stands for the minimum size of a colour class over all proper $\chi(H)$-colourings of $H$. We say that $G$ is $H$-good if $R(G,H)$ is equal to the general lower bound. Burr showed that, for any graph~$H$, every sufficiently long path is $H$-good. Our goal is to explore the notion of Ramsey goodness in the setting of $k$-uniform hypergraphs. We demonstrate that, in stark contrast to the graph case, $k$-uniform $\ell$-paths are not $H$-good for a large class of $k$-graphs. On the other hand, we prove that long loose paths are always at least asymptotically $H$-good for every $H$ and derive lower and upper bounds that are best possible in a certain sense. In the 3-uniform setting, we complement our negative result with a positive one, in which we determine the Ramsey number asymptotically for pairs containing a long tight path and a 3-graph $H$ when $H$ belongs to a certain family of hypergraphs. This extends a result of Balogh, Clemen, Skokan, and Wagner for the Fano plane asymptotically to a much larger family of 3-graphs.