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

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)≥(v(G)−1)(χ(H)−1)+σ(H), where σ(H) stands for the minimum size of a colour class over all proper χ(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 ℓ-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.