A note on 3‐partite graphs without 4‐cycles

Let C 4 be a cycle of order 4. Write e x ( n , n , n , C 4 ) 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 C 4 . In this paper, we show that e x ( n , n , n , C 4 ) ≥ 3 2 n ( p + 1 ) , where n = p ( p − 1 ) 2 and p is a prime number. Note that e x ( n , n , n , C 4 ) ≤(3 2 2 + o ( 1 ) ) n 3 2from 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 lim n → ∞e x ( n , n , n , C 4 )3 2 2 n 3 2= 1 .