[Paper Review] Linear Ramsey numbers for bounded-degree hypergraphs

Yoshiyasu Ishigami|ArXiv.org|Dec 20, 2006

Limits and Structures in Graph Theory23 references12 citations

TL;DR

This paper establishes that the Ramsey number of any bounded-degree $k$-uniform hypergraph is linear in the number of vertices, extending Chvátal et al.'s 1983 result for graphs. Using a novel hypergraph regularity lemma with a streamlined proof, the author constructs a counting lemma for blowups that enables embedding monochromatic copies of such hypergraphs in $b$-colored complete hypergraphs, proving $R_b(H) = O_b(N)$ for $k$-uniform hypergraphs with maximum degree $O(1)$.

ABSTRACT

We show that the Ramsey number is linear for every uniform hypergraph with bounded-degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. showed in 1983. Our proof is simple, contains the multicolor case, and provides a strong embedding lemma. It shows the potential of a new hypergraph regularity lemma by the author.

Motivation & Objective

• To extend the classical linear Ramsey number result for bounded-degree graphs to $k$-uniform hypergraphs.
• To establish that the Ramsey number $R_b(H)$ is linear in $N$ for any $k$-uniform hypergraph $H$ with maximum degree $O(1)$, for any fixed number of colors $b \geq 1$.
• To demonstrate the utility of a new hypergraph regularity lemma (from [18]) for Ramsey-theoretic problems.
• To provide a simpler, more modular proof framework that naturally accommodates multicolor Ramsey problems, unlike prior approaches.

Proposed method

• Applies a new hypergraph regularity lemma (Theorem 2.A) with a weak regularity setting, enabling shorter and more adaptable proofs.
• Uses a counting lemma for blowups (Corollary 2.3) that is stronger than prior embedding lemmas, relying on a carefully defined density threshold $\rho_i(b_i^*) = \alpha / b_i^*$.
• Constructs a regularity decomposition of the $b$-colored complete $k$-uniform hypergraph on $mN$ vertices into $m$ parts, each of size $N$, with controlled error parameters.
• Defines exceptional edges based on low density or high irregularity, showing their probability is bounded by $2\alpha$, and uses probabilistic method to find a vertex set avoiding them.
• Applies Ramsey's Theorem (Theorem 1.A) to find a monochromatic $k$-uniform subcomplex on $r = \Delta + 1$ vertices, then embeds the original hypergraph $B$ as a blowup of this complex.
• Employs a recursive embedding strategy via the main counting lemma, ensuring that the size constraints $|V_i(B)| < \eta_1(\rho_1(\widetilde{b}_1)) \cdot |V_i(\mathbf{G}^*)|$ are satisfied for the existence of the desired monochromatic copy.

Experimental results

Research questions

• RQ1 Does the linear Ramsey number property for bounded-degree graphs extend to $k$-uniform hypergraphs with maximum degree $O(1)$?
• RQ2 Can a new hypergraph regularity lemma framework (from [18]) be used to prove linear Ramsey numbers for hypergraphs more efficiently than existing methods?
• RQ3 Is the multicolor Ramsey problem for bounded-degree hypergraphs tractable using a regularity-based approach with a simpler proof structure?
• RQ4 Can the counting lemma for blowups in the new regularity setting be stronger than prior embedding lemmas in terms of generality and applicability?

Key findings

• The Ramsey number $R_b(H)$ is linear in $N$ for any $k$-uniform hypergraph $H$ on $N$ vertices with maximum degree $O(1)$, i.e., $R_b(H) = O_b(N)$.
• The proof relies on a new regularity lemma with a short, self-contained proof, contrasting with longer, less transparent proofs in prior works.
• The main counting lemma (Corollary 2.3) is stronger than the embedding lemmas in Cooley et al. and Nagle et al., as it applies to a weaker regularity setting.
• The method naturally extends to multicolor Ramsey problems from the start, unlike prior two-color-only approaches.
• The construction ensures that a monochromatic copy of $H$ exists in any $b$-edge-colored complete $k$-uniform hypergraph on $O(N)$ vertices, with the hidden constant depending only on $b$ and $k$.
• The result confirms the potential of the new regularity framework for broader applications in extremal combinatorics, as demonstrated in subsequent work [20].