[論文レビュー] An exponential improvement for diagonal Ramsey

Marcelo Campos, Simon Griffiths+2 more|arXiv (Cornell University)|Mar 16, 2023

Limits and Structures in Graph Theory被引用数 13

ひとこと要約

著者らは古典的な Erdős–Szekeres の上界に対して指数的改良を示し、対角ラムゼン数で R(k) ≤ (4−ε)^k を ε>0 のあることを示す、新しい Book Algorithm による。さらに ℓ ≤ k の領域でのオフ対角ラムゼン数 R(k,ℓ) の指数的改良を得る。

ABSTRACT

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

研究の動機と目的

• Motivate and bound Ramsey numbers by moving beyond the Erdős–Szekeres upper bound.
• Develop an elementary algorithmic approach that constructs large monochromatic books in colorings without large cliques.
• Translate the off-diagonal improvements into a stronger diagonal bound via a two-stage analysis.

提案手法

• Introduce the Book Algorithm that maintains disjoint sets (X, Y, A, B) with prescribed edge color properties.
• Use three moves: red steps, big blue steps, and density-boost steps to grow a red book (A, Y) while controlling red density p between X and Y.
• Define a weighted central-vertex strategy and a height-based step-size schedule to ensure p does not decrease too much across steps.
• Prove key lemmas (including a Zigzag Lemma) to bound the number and impact of steps, ensuring the algorithm yields large monochromatic books.
• Derive exponential upper bounds for R(k) via the off-diagonal bound R(k,ℓ) with ℓ ≈ k/5 and subsequently deduce the diagonal result.

実験結果

リサーチクエスチョン

• RQ1 Can we exceed the Erdős–Szekeres bound by exploiting a new algorithmic approach that constructs large monochromatic books?
• RQ2 How does the density-boost mechanism interact with red steps to maintain progress while controlling red edge density?
• RQ3 What exponential improvements to R(k,ℓ) can be achieved when ℓ ≤ k, and how do these feed into a diagonal bound for R(k)?

主な発見

• Prove that there exists ε>0 with R(k) ≤ (4−ε)^k for all sufficiently large k.
• Provide two concrete proofs of the diagonal bound with ε = 2^−10 and ε = 2^−7, indicating the constants can be improved further.
• Show an exponential improvement for off-diagonal Ramsey numbers: R(k,ℓ) ≤ e^{−δℓ+o(k)}{k+ℓ ￎill?}ℓ} for all k,ℓ with ℓ ≤ k (δ>0).
• Extend the method to bound R(k,ℓ) in the regime ℓ ≤ k/4 with the bound R(k,ℓ) ≤ e^{−ℓ/50+o(k)}{k+ℓ ￎill?}ℓ}.