Improved bounds for the minimum degree of minimal multicolor Ramsey graphs
Yamaan Attwa, Sam Mattheus, Tibor Szabó, Jacques Verstraete
TL;DR
This paper investigates the minimum degree of minimal $r$-Ramsey graphs for the clique $K_{k+1}$, focusing on upper and lower bounds and on semisaturation phenomena. It blends algebraic constructions from finite geometry with probabilistic refinements to produce $r$ edge-disjoint $K_{k+1}$-free graphs on a common vertex set that force a strong local density, yielding improved upper bounds for $s_r(K_k)$ in regimes where $k$ and $r$ grow with $k \le r \log^2 r$, namely $s_r(K_k) \le 2^{400} k^2 r^{2+30/k} \log^{20} r \log^{20} k$. The authors also derive upper bounds for colored semisaturation $\mathrm{ssat}_r(K_k)$ and establish a separation from $s_r(K_k)$, along with a nontrivial lower bound $s_r(K_k) \ge \Omega(k r^2)$, contributing to the understanding of the asymptotics across parameter ranges. The results leverage Hermitian unitals and projective geometry to guide the color pattern construction, with probabilistic tools ensuring the needed concentration properties, and they highlight a path toward tight $(rk)^{2+o(1)}$ upper bounds in broader regimes. The work also includes a semisaturation construction and open questions about the precise exponents and potential separations among related saturation notions in Ramsey theory.
Abstract
We provide two novel constructions of $r$ edge-disjoint $K_{k+1}$-free graphs on the same vertex set, each of which has the property that every small induced subgraph contains a complete graph on $k$ vertices. The main novelty of our argument is the combination of an algebraic and a probabilistic coloring scheme, which utilizes the beneficial algebraic and combinatorial properties of the Hermitian unital. These constructions improve on a number of upper bounds on the smallest possible minimum degree of minimal $r$-color Ramsey graphs for the clique $K_{k+1}$ when $r\geq c\frac{k}{\log^2 k}$ and $k$ is large enough.