An exponential improvement for diagonal Ramsey

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.

Figures (8)

• Figure 1: The sets $A$, $B$, $X$ and $Y$. All edges incident to $A$ are red, and all edges inside $B$ and between $B$ and $X$ are blue.
• Figure 2: We will show that if $n \geqslant (4 - o(1))^k$, then the pair $(x,y)$, where $x = t/k$ and $y = s/k$, lies in the blue region, and that if it lies outside the red region, then $|Y| \geqslant {2k - t \choose k-t}$. We are therefore happy unless $(x,y)$ lies in the intersection of the red and blue regions, which is coloured green.
• Figure 3: With our improved off-diagonal bound \ref{['eq:off:diag:just:enough']}, the red region (which corresponds to books $(A,Y)$ that are not big enough to find a monochromatic copy of $K_k$) becomes a little smaller, and the green region disappears.
• Figure 4: The sets $f(x,y) \geqslant 2$ (in red) and $g(x,y) \geqslant 2$ (in blue).
• Figure 5: The extreme cases of Lemmas \ref{['lem:nearish:calc']} and \ref{['lem:even:nearer:calc']}: roughly speaking, the statement of the lemmas is that the red and blue sets do not intersect.

Theorems & Definitions (125)

• Theorem 1.1
• Theorem 1.2
• Lemma 4.1
• proof : Proof of Lemma \ref{['lem:big:blue:step']}
• Lemma 4.3
• proof
• proof
• Lemma 5.1
• Lemma 5.2
• proof