Threshold Ramsey multiplicity for odd cycles


  • David Conlon Department of Mathematics, Caltech, Pasadena, CA 91125, USA
  • Jacob Fox Department of Mathematics, Stanford University, Stanford, CA 94305, USA
  • Benjamin Sudakov Department of Mathematics, ETH, 8092 Z¨urich, Switzerland
  • Fan Wei Department of Mathematics, Princeton University, Princeton, NJ 08540, USA



The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.


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