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New bounds on the generalized Ramsey number $f(n,5,8)$

Enrique Gomez-Leos, Emily Heath, Alex Parker, Coy Schwieder, Shira Zerbib

Abstract

Let $f(n,p,q)$ denote the minimum number of colors needed to color the edges of $K_n$ so that every copy of $K_p$ receives at least $q$ distinct colors. In this note, we show $\frac{6}{7}(n-1) \leq f(n,5,8) \leq n + o(n)$. The upper bound is proven using the "conflict-free hypergraph matchings method" which was recently used by Mubayi and Joos to prove $f(n,4,5) = \frac{5}{6}n + o(n)$.

New bounds on the generalized Ramsey number $f(n,5,8)$

Abstract

Let denote the minimum number of colors needed to color the edges of so that every copy of receives at least distinct colors. In this note, we show . The upper bound is proven using the "conflict-free hypergraph matchings method" which was recently used by Mubayi and Joos to prove .
Paper Structure (6 sections, 10 theorems, 33 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lower bound
  3. Conflict-free hypergraph matching method
  4. Upper bound
  5. Proof of Theorem \ref{['thm:coloringproperties']}
  6. Proof of Theorem \ref{['thm:upperbound']}

Key Result

Theorem 1

We have $f(n,5,8)\leq n+o(n)$.

Figures (1)

  • Figure 1: Bad subgraphs in $K_n$ which correspond to edges in the conflict hypergraph $\mathcal{C}$

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6: GJKKL, Theorem 3.3
  • Lemma 7: Lovász Local Lemma
  • ...and 4 more