Bounded VC-Dimension Implies the Schur-Erdos Conjecture

In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K-n, there is a monochromatic copy of K-3. He showed that r(3; m) <= O(m!), and a simple construction demonstrates that r(3; m) >= 2(omega(m)). An old conjecture of Erdos states that r(3; m) = 2(Theta(m)). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.