1,117 research outputs found
Chromatic Ramsey number of acyclic hypergraphs
Suppose that is an acyclic -uniform hypergraph, with . We
define the (-color) chromatic Ramsey number as the smallest
with the following property: if the edges of any -chromatic -uniform
hypergraph are colored with colors in any manner, there is a monochromatic
copy of . We observe that is well defined and where
is the -color Ramsey number of . We give linear upper bounds
for when T is a matching or star, proving that for , and where
and are, respectively, the -uniform matching and star with
edges.
The general bounds are improved for -uniform hypergraphs. We prove that
, extending a special case of Alon-Frankl-Lov\'asz' theorem.
We also prove that , which is sharp for . This is
a corollary of a more general result. We define as the 1-intersection
graph of , whose vertices represent hyperedges and whose edges represent
intersections of hyperedges in exactly one vertex. We prove that for any -uniform hypergraph (assuming ). The proof uses the list coloring version of Brooks' theorem.Comment: 10 page
The Set Multipartite Ramsey Numbers M_j(P_n, mK_2)
For given two any graph H and G, the set multipartite Ramsey number M_j(G, H) is the smallest integer t such that for every factorization of graph K_(t×j):= F1 F2 so that F1 contains G as a subgraph or F2 contains H as a subgraph. In this paper, we determine  M_j(P_n, mK_2) with j=3,4,5 and m>=2 where P_n denotes a path for n=2,3 vertices and mK_2 denotes a matching (stripes) of size m and pairwise disjoint edges
- …