Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers c, s ⩾ 1 , let M 3 ( c , s ) be the smallest integer such that any set of at least M 3 ( 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 M 3 ( 1 , 0 ) = 3 , M 3 ( 2 , 0 ) = 9 and M 3 ( c , 0 ) = ∞ , for c ⩾ 3 . In this paper, we prove that the smallest integer λ 3 ( c ) such that M 3 ( c , λ 3 ( c ) ) < ∞ is bounded above by c − 2 . We show this by proving M 3 ( c , c − 2 ) ⩽ ( c + 1 ) 2 , for c ⩾ 2 . We also determine the exact values of M 3 ( c , s ) for small values of c and s.