Upper bounds for multicolour Ramsey numbers (2410.17197v1)

Abstract: The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$, that $$R_r(k) \leqslant e^{-\delta k} r^{rk}$$ for some constant $\delta = \delta(r) > 0$ and all sufficiently large $k \in \mathbb{N}$. For each $r \geqslant 3$, this is the first exponential improvement over the upper bound of Erd\H{o}s and Szekeres from 1935. In the case $r = 2$, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe.

• The paper presents the first exponential improvement over classical bounds for r-color Ramsey numbers using a geometric lemma and density-boost methods.
• It introduces a novel algorithmic approach for efficiently constructing monochromatic cliques in multicolour edge colorings.
• The findings significantly enhance combinatorial understanding and open new avenues for research in graph theory and advanced Ramsey problems.

An Examination of Upper Bounds for Multicolour Ramsey Numbers

In the paper titled "Upper Bounds for Multicolour Ramsey Numbers," Balister et al. present novel results pertaining to bounding the multicolour Ramsey numbers more effectively than previous work. The ramification of these findings is substantial in the field of combinatorics, particularly in the paper of edge colorings in graphs.

The premise of the research hinges on the classical r-color Ramsey number, denoted as $R(k)_r$, which represents the smallest number $n$ such that any r-coloring of the edges of the complete graph $K_n$ contains a monochromatic clique of size $k$. Historically, the bound proposed by Erdős and Szekeres in 1935 had been one of the principal insights, dictating that $R(k)_r \leq r^{k/2}$. This paper presents the first exponential improvement over that bound for r greater than 3, marking a significant advancement.

The authors establish that for each fixed $r > 2$, there exists a constant $\delta = \delta(r) > 0$ such that:

$R(k)_r \leq e^{-\delta k} r^k$

for sufficiently large $k$. The paper's contribution is particularly notable since, for $r > 3$, improvements over the Erdős-Szekeres bound were previously nonexistent. In effectively lowering the upper bounds, the authors extend their approach beyond the case $r = 2$, which had been the focus of prior efforts.

The key methodology introduced in this paper revolves around a geometric lemma that highlights the behavior of negatively correlated functions defined over a finite set. These functions, under certain negative correlation conditions, must exhibit significant clustering, a concept that the authors leverage to construct monochromatic books more efficiently than prior techniques allowed.

Furthermore, the paper details a creative algorithmic strategy to identify these monochromatic books via density-boost techniques across multiple colors. This strategy involves partitioning vertex sets and applying local density conditions iteratively to enhance the clique sizes associated with a particular color.

Implications of these findings stretch across both theoretical explorations and practical applications within combinatorics. The exponential bound established enriches the understanding of how multicolour Ramsey numbers can be effectively constrained, thereby inviting further inquiry into more sophisticated coloring problems. The approach presented may also inspire analogous methods within other domains where combinatorial structures endure similar complexities.

Future research inspired by this paper may focus on refining the constants involved or exploring applications of the geometric lemma to other aspects of graph theory. The results may also drive interest in examining the interaction of combinatorial structures beyond the scope defined within this framework, particularly in areas necessitating the optimization of multicolour configurations.

In summary, this paper delivers a profound contribution to the paper of Ramsey theory by providing an effective exponential improvement on historical bounds for multicolour Ramsey numbers, thereby opening the field to further advancements and exploring new territories where such mathematical constructs theorize beyond previous limitations.