Multicolor Turán numbers II -- a generalization of the Ruzsa-Szemerédi theorem and new results on cliques and odd cycles

In this paper we continue the study of a natural generalization of Turán’s forbidden subgraph problem and the Ruzsa-Szemerédi problem. Let $e{x}_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. The case when both $F$ and $G$ are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when $F$ and $G$ are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear $r$-uniform hypergraph Turán problems to determine $e{x}_r^{lin}(n,G)$ form a class of the multicolor Turán problem, following the identity $e{x}_r^{lin}(n,G)=e{x}_{K_r}(n,G)$, our results determine the linear hypergraph Turán numbers of every graph of girth $3$ and for every $r$ up to a subpolynomial factor. Furthermore, when $G$ is a triangle, we settle the case $F=C_5$ and give bounds for the cases $F=C_{2k+1}$, $k\ge 3$ as well.