Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Ramsey Numbers in the Hypercube

Emily Heath, Coy Schwieder, Shira Zerbib

TL;DR

This work analyzes f(Q_n, C_{2k}, q) for the hypercube host, proving that for k≥3 and 3≤q≤k+1 the generalized Ramsey number satisfies f(Q_n, C_{2k}, q) ≤ c n^{(k−1)/(2k−q+1)} + n^{δ (k−1)/(2k−q+1)} with fixed 0<δ<1, by deploying the bipartite conflict-free matching method. The authors develop a detailed auxiliary bipartite hypergraph framework and a conflict system H, then verify degree, codegree, and 2-degree constraints to apply the Delcourt–Postle theorem, yielding colorings that ensure every C_{2k} receives at least q colors. They also derive lower bounds for f(Q_n, C_6, 4) and f(Q_n, C_6, 5) and provide a 4-coloring of Q_n guaranteeing that every C_4 is colored with at least 3 colors. The results extend prior Ramsey-type work on hypercubes and cycles, illustrating the effectiveness of conflict-free matching in structured graphs and suggesting avenues for broader H-subgraph generalizations and tighter exponents.

Abstract

We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gyárfás, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$.

Generalized Ramsey Numbers in the Hypercube

TL;DR

This work analyzes f(Q_n, C_{2k}, q) for the hypercube host, proving that for k≥3 and 3≤q≤k+1 the generalized Ramsey number satisfies f(Q_n, C_{2k}, q) ≤ c n^{(k−1)/(2k−q+1)} + n^{δ (k−1)/(2k−q+1)} with fixed 0<δ<1, by deploying the bipartite conflict-free matching method. The authors develop a detailed auxiliary bipartite hypergraph framework and a conflict system H, then verify degree, codegree, and 2-degree constraints to apply the Delcourt–Postle theorem, yielding colorings that ensure every C_{2k} receives at least q colors. They also derive lower bounds for f(Q_n, C_6, 4) and f(Q_n, C_6, 5) and provide a 4-coloring of Q_n guaranteeing that every C_4 is colored with at least 3 colors. The results extend prior Ramsey-type work on hypercubes and cycles, illustrating the effectiveness of conflict-free matching in structured graphs and suggesting avenues for broader H-subgraph generalizations and tighter exponents.

Abstract

We study the generalized Ramsey numbers , that is, the minimum number of colors needed to edge-color the hypercube so that every copy of the cycle has at least colors. Our main result is that for any integers satisfying and , we have We also prove a few other upper and lower bounds in the special cases and . This continues the line of research initiated by Faudree, Gyárfás, Lesniak, and Schelp and Mubayi and Stading who studied the case , and by Conder who considered the case and .
Paper Structure (9 sections, 7 theorems, 30 equations, 3 figures)

This paper contains 9 sections, 7 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The bipartite conflict-free matching method
  3. Proof of Theorem \ref{['thm: generalization']}
  4. Degree conditions of ${\mathcal{H}}$
  5. Codegree conditions
  6. Proof of Theorem \ref{['thm: 4 color C6']}
  7. Proof of Theorem \ref{['thm: 3 color C4']}
  8. Concluding remarks
  9. Acknowledgment

Key Result

Theorem 1

For any integer $k\ge 1$ such that $k\equiv 0 \bmod 4$, there exist constants $c_1, c_2$, depending only on $k$, such that In addition,

Figures (3)

  • Figure 1:
  • Figure 2: Color classes of $Q_n$.
  • Figure 3:

Theorems & Definitions (19)

  • Theorem 1: Mubayi-Stading MS
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: Delcourt-Postle DP
  • Theorem 6
  • Lemma 7
  • proof
  • proof : Proof of \ref{['thm: actualthm']}
  • Claim 8
  • ...and 9 more