Multi-color Ramsey number for trees
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For graphs \( G_i\), \( i=1,\dots,k\), let \( r(G_1,\dots,G_k)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete graph \( K_m\) are colored in \( k\) colors, then for some \( i\), \( 1\leq i\leq k\), there is a subgraph isomorphic to \( G_i\) with all edges in the \( i\)-th color.
A coloring problem for trees (proposed by Erdösand Graham [1])
Is it true for trees \(T_n\) on \(n\) vertices that \[ r(\underbrace{T_n, \ldots, T_n}_k) = kn +O(1)? \]This would follow from the Erdös-Sós conjecture on trees.
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1 |
P. Erdös,
Some new problems and results in graph theory and
other branches of combinatorial mathematics, Combinatorics and
graph theory (Calcutta, 1980), Lecture Notes in Math., 885,
9-17, Springer, Berlin-New York, 1981.
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