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 $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollobás conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ 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.
We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2×3 pattern is 1/720+o(1). This result is the best possible in the sense that 2×3 cannot be replaced with 2×2 or 1×n for any n.
We prove that in every 2-colouring of the edges of K N there exists a monochromatic infinite path P such that V (P ) has upper density at least (12 + √ 8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdős and Galvin.
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 r ε (H) such that for all n ≥ R r ε (H), all ε-balanced χ of Kn contain a subgraph isomorphic to H in its coloring. Bollobás conjectured a simple characterization of H for which R 2 ε (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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