Anti-Ramsey Numbers of Expansions of Doubly Edge-critical Graphs in Uniform Hypergraphs

For an r-graph H, the anti-Ramsey number ar (n, r, H) is the minimum number c of colors such that for any edge-coloring of the complete r-graph on n vertices with at least c colors, there is a copy of H whose edges have distinct colors. A 2-graph F is doubly edge-p-critical if the chromatic number (F - e) p for every edge e in F and there exist two edges e₁, e₂ in F such that (F -e₁- e₂) =p-1. The anti-Ramsey numbers of doubly edge-p-critical 2-graphs were determined by Jiang and Pikhurko Jiang&Pikhurko2009, which generalized the anti-Ramsey numbers of cliques determined by Erdos, Simonovits and S\'os Erdos&Simonovits&Sos1975. In general, few exact values of anti-Ramsey numbers of r-graphs are known for r 3. Given a 2-graph F, the expansion F^ (r) of F is an r-graph on |V (F) |+ (r-2) |F| vertices obtained from F by adding r-2 new vertices to each edge of F. In this paper, we determine the exact value of ar (n, r, F^ (r) ) for any doubly edge-p-critical 2-graph F with p>r 3 and sufficiently large n.