28≤R(C4,C4,C3,C3)≤36

Abstract

Using four colors we construct a coloring of the edges of K₂₇ which has no monochromatic quadrilaterals in the first two colors and no monochromatic triangles in the other two colors. This gives a new lower bound of 28 on the Ramsey number R(C₄,C₄,C₃,C ₃). We also prove an upper bound of 36 for the same number using an estimate of the maximum number of edges in C₄-free graphs.