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A linear upper bound on the zero-sum Ramsey number of forests in $\mathbb{Z}_p$

Lucas Colucci, Marco D'Emidio

Abstract

Let $m$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,\mathbb{Z}_m)$ is the least integer $N$ (if it exists) such that for every edge-coloring $χ\, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_m$ one can find a copy of $G$ in $K_N$ such that $\sum_{e \, \in \, E(G)}{χ(e)} \, = \, 0$. In this paper, we show that, for every prime $p$, $$R(F,\mathbb{Z}_p)\leq n+9p-12$$ for every forest $F$ in $n\geq 3p^2-12p+11$ vertices with $p\mid e(F)$ without isolated vertices.

A linear upper bound on the zero-sum Ramsey number of forests in $\mathbb{Z}_p$

Abstract

Let be a positive integer and let be a graph. The zero-sum Ramsey number is the least integer (if it exists) such that for every edge-coloring one can find a copy of in such that . In this paper, we show that, for every prime , for every forest in vertices with without isolated vertices.

Paper Structure

This paper contains 10 sections, 13 theorems, 28 equations.

Table of Contents

  1. Introduction and main result
  2. Definitions and notation
  3. Proof of the main result
  4. Bushy forests
  5. The coloring $\chi$ is vibrant
  6. The case when $\chi$ is non-vibrant
  7. Non-bushy forests
  8. The coloring $\chi$ is switchable
  9. The coloring $\chi$ is non-switchable
  10. Conclusion and open questions

Key Result

Theorem 1

Let $k$ be an integer and suppose $k \mid \binom{n}{2}$.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5: Bushy forest
  • Definition 6: $b$-colorful vertex
  • Definition 7: Vibrant coloring
  • Definition 8: Switcher $C_4$
  • Definition 9: Switchable coloring
  • Theorem 10: Generalized Cauchy-Davenport
  • ...and 25 more