Density of Balanced 3-Partite Graphs without 3-Cycles or 4-Cycles

Abstract

Let $C_k$ be a cycle of order $k$, where $k\ge 3$. Let ex$(n, n, n, \{C_{3}, C_{4}\})$ be the maximum number of edges in a balanced $3$-partite graph whose vertex set consists of three parts, each has $n$ vertices that has no subgraph isomorphic to $C_3$ or $C_4$. We construct dense balanced 3-partite graphs without 3-cycles or 4-cycles and show that ex$(n, n, n, \{C_{3}, C_{4}\})\ge (\frac{6\sqrt{2}-8}{(\sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.