This paper extends to hypergraphs a question discussed in~Bialostocki and Gyárfás  and Garrison : can one generalize a Ramsey type result from complete host graphs to graphs of sufficiently large chromatic number? For an r-uniform tree T (r≥2) we define the (t-color) chromatic Ramsey number χ(T,t)as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T. The presence of a tree is not accidental: χ(H,t) can be defined only for an acyclic hypergraph H since there are hypergraphs with arbitrary large chromatic number and girth. We prove that +1≤χ(T,t)≤|E(T)|t+1 where Rr(T,t) is the t-color Ramsey number of T. We give better upper bounds for χ(T,t) when T is a matching or a star, proving that for r≥2,k≥1,t≥1, χ(Mk r,t)≤(t−1)(k−1)+2k and χ(Sk r,t)≤t(k−1)+2 where Mk r and Sk r are, respectively, the r-uniform matching and star with k edges. The general upper bounds are improved for 3-uniform hypergraphs. We prove that χ(Mk 3,2)=2k, extending a special case of Alon–Frankl–Lovász theorem. We also prove that χ(S2 3,t)≤t+1, which is sharp for t=2,3. This is a corollary of our main result which bounds the chromatic number χ(H) of 3-uniform hypergraphs by the chromatic number of its 1-intersection graph H, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that χ(H)≤χ(H) for any 3-uniform hypergraph H (assuming that H has at least one edge). The proof uses the list coloring version of Brooks’ theorem. The more general question, whether χ(H)≤χ(H) holds for every r-uniform hypergraph (r>3) remains open. We could not decide either whether the above lower bound of χ(T,t) is sharp for every r-uniform tree.
- Chromatic number of triple systems
- Ramsey number of matchings
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics