Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles

• Published in: Journal of graph theory Vol. 107; no. 3; pp. 629 - 641
• Main Authors: Kovács, Benedek, Nagy, Zoltán Lóránt
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.11.2024
• Subjects: extremal graphs; graph packings; Graph theory; Graphs; hypergraph Turán problems; multicolor; Theorems
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23147

In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let ex F ( n , G ) ${\text{ex}}_{F}(n,G)$ denote the maximum number of edge‐disjoint copies of a fixed simple graph F $F$ that can be placed on an n $n$‐vertex ground set without forming a subgraph G $G$ whose edges are from different F $F$‐copies. The case when both F $F$ and G $G$ are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when F $F$ and G $G$ 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 $r$‐uniform hypergraph Turán problems to determine ex r l i n ( n , G ) ${\text{ex}}_{r}^{lin}(n,G)$ form a class of the multicolor Turán problem, following the identity ex r l i n ( n , G ) = ex K r ( n , G ) ${\text{ex}}_{r}^{lin}(n,G)={\text{ex}}_{{K}_{r}}(n,G)$, our results determine the linear hypergraph Turán numbers of every graph of girth 3 and for every r $r$ up to a subpolynomial factor. Furthermore, when G $G$ is a triangle, we settle the case F = C 5 $F={C}_{5}$ and give bounds for the cases F = C 2 k + 1 $F={C}_{2k+1}$, k ≥ 3 $k\ge 3$ as well.