Finding unavoidable colorful patterns in multicolored graphs
release_auoxvch6hbbd3ap6mbyatdlqsu
by
Matthew Bowen and Ander Lamaison and Necati Alp Müyesser
2018
Abstract
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 χ of K_n contain a 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, discuss
quantitative bounds, as well as generalizations to infinite graphs.
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