LAUSR

Let C4 be a cycle of order 4. Write ex(n,n,n,C4) for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has n vertices that have no subgraph isomorphic to C4 . In this paper, we show that ex(n,n,n,C4)≥32n(p+1) , where n=p(p−1)2 and p is a prime number. Note that ex(n,n,n,C4)≤(322+o(1))n32 from Tait and Timmons's works. Since for every integer m , one can find a prime p such that m≤p≤(1+o(1))m , we obtain that limn→∞ex(n,n,n,C4)322n32=1 .