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Title:
Ramsey numbers of degenerate graphs
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Abstract:
The Ramsey number of a graph $G$ is the minimum integer $n$ for which every edge-coloring of the complete graph on $n$ vertices with two colors contains a monochromatic copy of $G$. A graph is $d$-degenerate if all its subgraphs have a vertex of degree at most $d$. In this talk, we prove that for all $d$, there exists a constant $c$ such that every $d$-degenerate graph $G$ has Ramsey number at most $c|V(G)|$. This solves a conjecture of Burr and Erdős from 1973.
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