5 research outputs found
Ramsey Problems for Berge Hypergraphs
For a graph G, a hypergraph is a Berge copy of G (or a Berge-G in short) if there is a bijection such that for each we have . We denote the family of r-uniform hypergraphs that are Berge copies of G by . For families of r-uniform hypergraphs and , we denote by the smallest number n such that in any red-blue coloring of the (hyper)edges of (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in or a monochromatic red copy of a hypergraph in . denotes the smallest number n such that in any coloring of the hyperedges of with c colors, there is a monochromatic copy of a hypergraph in . In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if , then . In the case r = 2c, we show that , and if G is a noncomplete graph on n vertices, then , assuming n is large enough. In the case we also obtain bounds on . Moreover, we also determine the exact value of for every pair of trees T_1 and T_2.
Read More: https://epubs.siam.org/doi/abs/10.1137/18M1225227?journalCode=sjdme
Large monochromatic components in colorings of complete 3-uniform hypergraphs
AbstractLet f(n,r) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph Kn3 are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that f(n,5)≥5n7 and f(n,6)≥2n3. Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively)