The Ramsey number of the Fano plane versus the tight path

The hypergraph Ramsey number of two 3-uniform hypergraphs G and H, denoted by R(G, H), is the least integer N such that every red-blue edge-coloring of the complete 3-uniform hypergraph on N vertices contains a red copy of G or a blue copy of H. The Fano plane F is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that R(G, F) >= 2(v(G) -1) +1 for every connected G. Hypergraphs G for which the equality R(G, F) = 2(v(G) - 1) + 1 holds are called F-good. Conlon posed the problem to determine all G that are F-good. In this short paper we make progress on this problem by proving that the tight path of length n is F-good.