Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers c, s⩾1, let M3(c,s) be the smallest integer such that any set of at least M3(c,s) points in the plane, no three on a line, and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s=0, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3(1,0)=3, M3(2,0)=9 and M3(c,0)=∞, for c⩾3. In this paper, we prove that the smallest integer λ3(c) such that M3(c,λ3(c))<∞ is bounded above by c−2. We show this by proving M3(c,c−2)⩽(c+1)2, for c⩾2. We also determine the exact values of M3(c,s) for small values of c and s.