# Generalizing the Ramsey Problem through Diameter

@article{Mubayi2002GeneralizingTR,
title={Generalizing the Ramsey Problem through Diameter},
author={Dhruv Mubayi},
journal={Electron. J. Comb.},
year={2002},
volume={9}
}
• D. Mubayi
• Published 13 November 2001
• Mathematics, Computer Science
• Electron. J. Comb.
Given a graph $G$ and positive integers $d,k$, let $f_d^k(G)$ be the maximum $t$ such that every $k$-coloring of $E(G)$ yields a monochromatic subgraph with diameter at most $d$ on at least $t$ vertices. Determining $f_1^k(K_n)$ is equivalent to determining classical Ramsey numbers for multicolorings. Our results include $\bullet$ determining $f_d^k(K_{a,b})$ within 1 for all $d,k,a,b$ $\bullet$ for $d \ge 4$, $f_d^3(K_n)=\lceil n/2 \rceil +1$ or $\lceil n/2 \rceil$ depending on whether $n… ## Topics from this paper Monochromatic balanced components, matchings, and paths in multicolored complete bipartite graphs • Mathematics • 2018 It is well-known that in every$r$-coloring of the edges of the complete bipartite graph$K_{n,n}$there is a monochromatic connected component with at least${2n\over r}$vertices. It would be Large monochromatic components in multicolored bipartite graphs • Mathematics, Computer Science J. Graph Theory • 2020 The conjecture that in every$r$-coloring of the edges of the complete bipartite graph there is a monochromatic connected component with at least${m+n\over r}$vertices is conjecture and a weaker bound for all$r\geq 3$is proved. Long monochromatic paths and cycles in 2-colored bipartite graphs • Computer Science, Mathematics Discret. Math. • 2020 Partitioning Random Graphs into Monochromatic Components • Mathematics, Computer Science Electron. J. Comb. • 2017 This work extends Haxell and Kohayakawa's result to graphs with large minimum degree, then provides some partial analogs of their result for random graphs and considers a random graph version of a classic result of Gyarfas (1977) about large monochromatic components in$r-colored complete graphs.
Monochromatic Partitions In Local Edge Colorings
An edge coloring of a graph is a local r -coloring if the edges incident to any vertex are colored with at most r distinct colors. In this paper, generalizing our earlier work, we study the following
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• Mathematics
• 2015
We propose a variant of tree cover numbers for edge coloured graphs. More precisely, we consider graphs whose edges are each coloured with a set of $k$ colours, where the total number of colours is
Size of Monochromatic Double Stars in Edge Colorings
• Mathematics, Computer Science
Graphs Comb.
• 2008
It is shown that in every r-coloring of the edges of Kn there is a monochromatic double star with at least n(r+1)+r-1}{r^2+1} vertices, which improves a bound of Mubayi for the largest monochromeatic subgraph of diameter at most three.
Ramsey-type results for Gallai colorings
• Computer Science
J. Graph Theory
• 2010
It is proved that in every G-coloring of Kn there exists each of the following: a monochromatic double star with at least 3n+1 4 vertices; and RG(r,K3) can be determined exactly.
The Erd\H{o}s-Hajnal conjecture for three colors and multiple forbidden patterns
• Mathematics
• 2020
Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n)

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