28≤R(C4,C4,C3,C3)≤36 - Xiaodong, Xu; Radziszowski, Stanislaw P. (Utilitas Mathematica, 2009)
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.