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 χ(F - e)≥ p for every edge e in F and there exist two edges e₁,e₂ in F such that χ(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 Erd{o}s, Simonovits and S\'{o}s {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.