[Paper Review] Ramsey numbers of cubes versus cliques

David Conlon, Jacob Fox+2 more|arXiv (Cornell University)|Aug 8, 2012

Limits and Structures in Graph Theory23 references4 citations

TL;DR

This paper establishes the first upper bound on the Ramsey number $ r(Q_n, K_s) $ that is within a constant factor of the known lower bound $ (s-1)(2^n - 1) + 1 $, proving that $ r(Q_n, K_s) \leq c_s 2^n $ for some constant $ c_s $ depending on $ s $. The result resolves a long-standing conjecture by Burr and Erd\'os on the $ s $-goodness of hypercubes, showing that the cube graph $ Q_n $ is $ s $-good for fixed $ s \geq 3 $ and sufficiently large $ n $, using techniques from graph decomposition and separator theorems.

ABSTRACT

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.

Motivation & Objective

• To resolve a conjecture by Burr and Erd\'os on whether the hypercube $ Q_n $ is $ s $-good for fixed $ s \geq 3 $ and large $ n $.
• To close the gap between the known lower bound $ (s-1)(2^n - 1) + 1 $ and the best previous upper bound for $ r(Q_n, K_s) $.
• To establish that the Ramsey number of the $ n $-dimensional cube versus a clique of size $ s $ is linear in $ 2^n $, up to a constant factor.
• To apply graph separator theorems and degeneracy-based embedding techniques to hypercubes, which lack bounded degree and thus evade prior methods.

Proposed method

• Uses a recursive decomposition of the graph into sparse subgraphs with controlled expansion properties, relying on a $(|V|^{1-\gamma}, \eta)$-separator structure.
• Applies the result of Nikiforov and Rousseau that $ d $-degenerate graphs with small separators are $ K $-good for $ K \in \mathcal{K} $, where $ \mathcal{K} $ includes graphs with at least two singleton color classes in a $ \chi(K) $-coloring.
• Employs iterative refinement of vertex sets to construct a hierarchy of subgraphs with decreasing size and increasing sparsity, ensuring no edges between certain parts.
• Establishes that $ Q_n $-free graphs with high minimum degree must contain a large independent set, using extremal graph theory and Turán-type bounds.
• Leverages the fact that $ Q_n $ has sublinear bandwidth and is bipartite, but not of bounded degree, requiring new techniques beyond previous $ H $-goodness results.
• Uses dependent random choice and probabilistic embedding lemmas to control the structure of graphs avoiding $ Q_n $, enabling the construction of large independent sets.

Experimental results

Research questions

• RQ1 Is the hypercube $ Q_n $ $ s $-good for fixed $ s \geq 3 $ and sufficiently large $ n $, meaning $ r(Q_n, K_s) = (s-1)(2^n - 1) + 1 $?
• RQ2 Can the upper bound on $ r(Q_n, K_s) $ be improved to within a constant factor of the known lower bound?
• RQ3 Do graphs with sublinear bandwidth and high minimum degree necessarily contain $ Q_n $ or have large independent sets?
• RQ4 Can separator theorems and degeneracy conditions be used to prove $ H $-goodness for graphs like $ Q_n $ that are not of bounded degree?

Key findings

• The Ramsey number $ r(Q_n, K_s) $ satisfies $ r(Q_n, K_s) \leq c_s 2^n $ for some constant $ c_s $ depending only on $ s $, matching the lower bound up to a constant factor.
• This establishes that the family of hypercubes $ \{Q_n\} $ is $ s $-good for every fixed $ s \geq 3 $, confirming a conjecture of Burr and Erd\'os.
• The result implies that for any fixed graph $ H $, $ r(Q_n, H) \leq c_H 2^n $, showing $ Q_n $ is $ H $-good for all fixed $ H $.
• The proof relies on a novel application of separator theorems to $ d $-degenerate graphs with small separators, extending known $ K $-goodness results to unbounded-degree graphs.
• The authors construct a hierarchy of vertex sets with decreasing size and increasing sparsity, ensuring no edges between certain components, which enables the embedding of $ Q_n $ or the detection of large independent sets.
• The work provides a foundational step toward resolving the broader Burr-Erd\'os conjecture on the linearity of $ r(Q_n, Q_n) $, as a positive answer would imply $ r(Q_n, K_s) $ is linear in $ 2^n $.