2023/12/11 by Jacob Fox, Fox, Jacob, Jonathan Tidor +3
Computer Science · Mathematics · #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.2312.06895
openalex publication_date 2023/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A graph is H-Ramsey if every two-coloring of its edges contains a monochromatic copy of H. Define the F-Ramsey number of H, denoted by rF(H), to be the minimum number of copies of F in a graph which is H-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that rK3(Kt)=\binomr(Kt)3 for all sufficiently large t. We do so through a result on graph coloring: there exists an absolute constant K such that every r-chromatic graph where every edge is contained in at least K triangles must contain at least \binomr3 triangles in total.