Finding unavoidable colorful patterns in multicolored graphs [article]

Matthew Bowen and Ander Lamaison and Alp Müyesser
2020 arXiv   pre-print
We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of K_n where each color is well-represented. Let χ be a coloring of the edges of a complete graph on n vertices into r colors. We call χ ε-balanced if all color classes have ε fraction of the edges. Fix some graph H, together with an r-coloring of its edges. Consider the smallest natural number R_ε^r(H) such that for all n≥ R_ε^r(H), all ε-balanced colorings χ of K_n contain a
more » ... subgraph isomorphic to H in its coloring. Bollobás conjectured a simple characterization of H for which R_ε^2(H) is finite, which was later proved by Cutler and Montágh. Here, we obtain a characterization for arbitrary values of r, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre.
arXiv:1807.02780v3 fatcat:d7e35j5livdzlfcks33fdbbk4m