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Multicolor, multipartite Ramsey numbers for quadrilateral

Janusz Dybizbański, Yaser Rowshan

Abstract

The $p$-partite Ramsey number for quadrilateral, denoted by $r_p(C_4,k)$, is the least positive integer $n$ such that any coloring of the edges of a complete $p$-partite graph with $n$ vertices in each partition with $k$ colors will result in a monochromatic copy of $C_4$. In this paper, we present an upper bound for $r_p(C_4,k)$ and the exact values of $r_p(C_4,2)$ for all $p\geq2$. In tripartite case we show that $r_3(C_4,k) \leq \lfloor (k+1)^2/2\rfloor-1$ and the exact value of 4-color tripartite Ramsey number $r_3(C_4,4)=11$.

Multicolor, multipartite Ramsey numbers for quadrilateral

Abstract

The -partite Ramsey number for quadrilateral, denoted by , is the least positive integer such that any coloring of the edges of a complete -partite graph with vertices in each partition with colors will result in a monochromatic copy of . In this paper, we present an upper bound for and the exact values of for all . In tripartite case we show that and the exact value of 4-color tripartite Ramsey number .
Paper Structure (4 sections, 6 theorems, 13 equations, 1 figure, 1 table)

This paper contains 4 sections, 6 theorems, 13 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Upper bound
  3. Two colors multipartite Ramsey numbers
  4. Tripartite Ramsey numbers

Key Result

Theorem 1

Let $a_1,...,a_n$ be a sequence of non-negative integers and $M=\sum_{i=1}^n a_i$, then where $a$ and $r$ are integers such that $M=an+r$ and $0\leq r < n$. Moreover, the minimum is reachable if and only if $|a_i-a_j|\leq 1$ for every $1\leq i < j \leq n$.

Figures (1)

  • Figure 1: Matrix of $4$-edge-coloring of $K_{10}^3$ without monochromatic $C_4$.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Claim 5
  • proof
  • Claim 6
  • ...and 5 more