Bounds for two multicolor Ramsey numbers concerning quadrilaterals

For k given graphs H₁,…,H_k, k≥2, the k-color Ramsey number, denoted by R(H₁,…,H_k), is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of H_i colored with i, for some 1≤i≤k. Let C_m be a cycle of length m, K_1,n a star of order n+1 and W_n a wheel of order n+1. In this paper, by using algebraic and probabilistic methods, we first give two lower bounds for (k+1)-color Ramsey number R(C₄,…,C₄,K_1,n) for some special n, which shows the upper bound due to Zhang et al. (2019) is tight in some sense, and then establish a general lower bound for R(C₄,…,C₄,K_1,n) in terms of n and k, which extends the classical result of Burr et al. (1989). Moreover, we show that R(C₄,…,C₄,K_1,n)=R(C₄,…,C₄,W_n) for sufficiently large n.