2025/02/17 by Li, Ping
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.11869
The Ruzsa-Szemerédi (6,3)-problem can be equivalently stated as determining the maximum number of edge-disjoint triangles on n vertices such that no triangle is formed by edges from three distinct triangle-copies. Gowers and Janzer extended this problem by establishing an analogous result for complete graphs. A natural generalization of the two results, first introduced by Imolay, Karl, Nazy and Váli, asks for the maximum number of edge-disjoint copies of a graph F on n vertices such that no copy of G is formed by edges originating from distinct F-copies. This maximum number, denoted by exF(n,G), is called the \em F-multicolor Turán number of G. This paper focuses on the setting of uniform hypergraphs. We first prove that for k-uniform hypergraphs G and F, exF(n,G)=o(nk) if and only if there exists a homomorphism from G to F. For degenerate case, we show that exF(n,G)=nk-o(1) whenever G contains a k-uniform tight triangle. These results extend previous results. We further establish corresponding supersaturation and blowup statements. In the non-degenerate setting, we derive matching lower and upper bounds for exF(n,G). We give a necessary and sufficient condition for exF(n,G) to fail to attain the upper bound, under the assumption that the extremal graphs for G are stable. As an application, we refine a result due to Imolay, Karl, Nazy and Váli. Furthermore, we completely characterize F for which exF(n,G) does not attain the upper bound when G is one of the three special intersecting graphs: Fano plane, extended triangle and r-book of r-edges with r=3,4.