[Paper Review] A generalization of Ramsey theory for stars and one matching

Amir Khamseh, Gholamreza Omidi|arXiv (Cornell University)|Mar 11, 2012

Limits and Structures in Graph Theory2 references3 citations

TL;DR

This paper generalizes Ramsey-type problems by determining the minimum number of vertices required in a t-edge-colored complete graph to guarantee either s monochromatic stars or one matching, using extremal graph theory and Vizing's theorem. The key result provides exact formulas for the d-chromatic Ramsey number r_{t-1}^t for configurations involving stars and one matching under various conditions on s and the star sizes.

ABSTRACT

A recent question in generalized Ramsey theory is that for fixed positive integers $s\leq t$, at least how many vertices can be covered by the vertices of no more than $s$ monochromatic members of the family $\cal F$ in every edge coloring of $K_n$ with $t$ colors. This is related to {$d$-chromatic Ramsey numbers} introduced by Chung and Liu. In this paper, we first compute these numbers for stars generalizing the well-known result of Burr and Roberts. Then we extend a result of Cockayne and Lorimer to compute $d$-chromatic Ramsey numbers for stars and one matching.

Motivation & Objective

• To generalize classical Ramsey theory by extending results on monochromatic stars and matchings to t-color edge-colored complete graphs.
• To compute the d-chromatic Ramsey number r_{t-1}^t(G_1,…,G_t) for families including stars and one matching, where d = t-1.
• To establish tight upper and lower bounds for the Ramsey number under different constraints on the size of the matching and star sizes.
• To unify and extend prior results by Burr, Roberts, Cockayne, and Lorimer in the context of generalized Ramsey numbers.

Proposed method

• Uses extremal graph theory to bound the number of edges avoiding a star K_{1,m_i}, leveraging the inequality ex(p, K_{1,m}) ≤ p(m-1)/2.
• Applies a coloring argument based on the pigeonhole principle: if the sum of (m_i - 1) over all colors is large, then one color class must contain a star in the remaining t-1 colors.
• Employs Vizing’s theorem to construct edge-colorings of subgraphs with t-1 colors, ensuring controlled degrees and avoiding monochromatic stars.
• Constructs explicit colorings of K_n using partitioning of vertex sets and careful assignment of colors to edges to achieve lower bounds.
• Uses induction and recursive reduction by relating r_{t-1}^t to r_{t-2}^{t-1} to simplify the problem to smaller cases.
• Analyzes maximal matchings in the union of t-1 colors to derive contradictions when vertex degrees exceed thresholds, proving existence of required subgraphs.

Experimental results

Research questions

• RQ1 What is the minimum number of vertices required in a t-edge-colored complete graph to guarantee s monochromatic stars or one matching in the union of t-1 colors?
• RQ2 How does the d-chromatic Ramsey number r_{t-1}^t(K_{1,m_1},…,K_{1,m_t}) behave when one of the graphs is replaced by a matching instead of a star?
• RQ3 Under what conditions does the Ramsey number equal 2s, R_{t-1}, or (⌈(∑ + s)/(t-1)⌉) + 1?
• RQ4 How do the bounds depend on the total size of the stars and the size of the matching?
• RQ5 Can the generalized Ramsey number be exactly determined for t-colorings when d = t-1 and the family includes stars and one matching?

Key findings

• The d-chromatic Ramsey number r_{t-1}^t for t-1 stars and one matching is exactly R = R_{t-1} when 2s ≥ R_{t-1}, where R_{t-1} is the Ramsey number for t-1 stars in t-1 colors.
• When 2s < R_{t-1} and the sum of (m_i - 1) over i is less than (2t - 3)s - t + 2, the Ramsey number is exactly 2s.
• When 2s < R_{t-1} and ∑ ≥ (2t - 3)s - t + 2, the Ramsey number is R = ⌈(∑ + s)/(t - 1)⌉ + 1.
• The paper provides a complete classification of the Ramsey number for t-colorings with d = t-1, depending on the relative sizes of s and the star sizes.
• The results generalize known results for stars (d=1) and extend the Cockayne-Lorimer theorem to t-colorings with d = t-1.
• The construction of extremal colorings using Vizing’s theorem and vertex partitioning proves the tightness of the upper bounds, establishing exact values.