Tight path, what is it (Ramsey-) good for? Absolutely (almost) nothing!

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, the Ramsey number R(G;H) is bounded below by (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. Together with Erdõs, he then asked when this lower bound is attained, introducing the notion of Ramsey goodness and its systematic study. We say that G is H-good if the Ramsey number of (G;H) is equal to the general lower bound. Among other results, it was shown by Burr 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 3-uniform hypergraphs. Motivated by Burr’s result concerning paths and a recent result of Balogh, Clemen, Skokan, and Wagner, we ask: what 3-graphs H is a (long) tight path good for? We demonstrate that, in stark contrast to the graph case, long tight paths are generally not H-good for various types of 3-graphs H. Even more, we show that the ratio R(P_n, H)/n for a pair (P_n, H) consisting of a tight path on n vertices and a 3-graph H cannot in general be bounded above by any function depending only on χ(H). We complement these negative results with a positive one, determining the Ramsey number asymptotically for pairs (P_n, H) when H belongs to a certain family of hypergraphs.