2018/01/01 by Jessica De Silva, Xiang Si, De Silva, Jessica +9
Computer Science · Mathematics · #05A16 #05C15 #05D10 #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1801.00474
openalex publication_date 2018/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Ramsey multiplicity constant of a graph H is the minimum proportion of copies of H in the complete graph which are monochromatic under an edge-coloring of Kn as n goes to infinity. Graphs for which this minimum is asymptotically achieved by taking a random coloring are called \em common, and common graphs have been studied extensively, leading to the Burr-Rosta conjecture and Sidorenko's conjecture. Erdős and Sós asked what the maximum number of rainbow triangles is in a 3-coloring of the edge set of Kn, a rainbow version of the Ramsey multiplicity question. A graph H is called r-anti-common if the maximum proportion of rainbow copies of H in any r-coloring of E(Kn) is asymptotically achieved by taking a random coloring. In this paper, we investigate anti-Ramsey multiplicity for several families of graphs. We determine classes of graphs which are either anti-common or not. Some of these classes follow the same behavior as the monochromatic case, but some of them do not. In particular the rainbow equivalent of Sidorenko's conjecture, that all bipartite graphs are anti-common, is false.