For every fixed integer $t \geq 2$, we give an upper bound on the multicolor vector space Ramsey number $R_2(t; k)$ that is a tower function of height independent of $k$. For $t \geq 3$, this is the first bound of its form, significantly improving upon the earlier bounds that are towers of height linear in $k$. We achieve this by reducing the problem to a classical hypergraph Ramsey problem via binary simplex codes. In particular, we prove that $$R_2(t; k) \leq \left\lceil \log R(K_s^{(r)}; k + 1) \right\rceil \leq \mathrm{twr}_{r-1}(c k\log k),$$ for $r = 2^{t - 1}$ and $s = 2^t - 1$, where $R(K_{s}^{(r)}; k + 1)$ is the classical $(k + 1)$-color Ramsey number for the complete $r$-uniform hypergraph on $s$ vertices. This improvement also translates into an improved lower bound on the chromatic number of the binary projective space with respect to $(t - 1)$-flats. For $t = 2$, it recovers the connection with multicolor Ramsey numbers for triangles.
In a recent breakthrough, OpenAI proved that the $k$-color Ramsey number of the triangle $C_3$ grows super-exponentially, more precisely, they proved that $R_k(C_3)\ge k^{k/3-o(k)}$. In this short note, we present a modification of their recursive construction that works for multicolor Ramsey numbers of fixed odd cycle...
Let $R_r(k)$ denote the diagonal $r$-color graph Ramsey number. We prove that there exist absolute constants $c,K>0$ such that \[ R_r(k)\le \exp\!\left(-c\frac{k}{r^2\log^4(2r)}\right)r^{rk} \] for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. The proof combines a positive-coefficient root filter of variable order wit...
Let $R_r(k)$ denote the diagonal $r$-colour Ramsey number. We prove that there exist absolute constants $c,K>0$ such that $R_r(k)\le r^{rk}\exp\!\left(-c\frac{k}{r\log^2(2r)}\right)$ for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. This improves the exponential saving in a recent bound of Yang and Mao by a factor of...
We provide an explicit algebraic construction showing that, uniformly for integers $3 \leq s \leq t$, as $t \to \infty$, \[ R( s,t ) \geq t^{(1-o(1)) \log s / \log(\log s + 1) }. \] For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudl\'ak. In particular, $R(...
The \emph{$k$-color Ramsey number} $R_k(C_{2\ell+1})$ is the least integer $n$ such that any $k$-edge-coloring of a complete graph $K_n$ has a monochromatic odd cycle $C_{2\ell+1}$. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundstr\"om~(JCT-B, 2026) recently proved \[ R_k(C_{2\ell+1})\le (4\ell-2)^k k^{k/\ell...
For positive integers $n,r,t$, let $\delta(n,r,t)$ denote the maximum possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and chromatic number at most $t$. Lo, Treglown and Zhao established a general upper bound for this parameter and used it, together with explicit constructions, to determin...
Yu Kasugai· 0 citations
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