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 ${\text{ex}}_F\left(n,G\right)$ 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 ${\text{ex}}_r^{lin}\left(n,G\right)$ form a class of the multicolor Turán problem, following the identity ${\text{ex}}_r^{lin}\left(n,G\right)={\text{ex}}_{K_r}\left(n,G\right)$, 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.