Multi-color Ramsey number for trees

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.

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.