Bounds on Some Ramsey Numbers Involving Quadrilateral

Abstract

For graphs \(G_1, G_2, \ldots, G_m\), the Ramsey number \(R(G_1, G_2, \ldots, G_m)\) is defined to be the smallest integer \(n\) such that any \(m\)-coloring of the edges of the complete graph \(K_n\) must include a monochromatic \(G_i\) in color \(i\), for some \(i\). In this note, we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral \(C_4\), including:\(R(C_4, K_9) \leq 32,
19 \leq R(C_4, C_4, K_4)\leq 22, 31 \leq R(C_4, C_4, C_4, K_4) \leq 50, 52 \leq R(C_4, K_4, K_4) \leq 72, 42 \leq R(C_4, C_4, K_3, K_5) \leq 76, 87\leq R(C_4, C_4, K_4, K_4) \leq 179.\)