AI 中文总结
研究二元域上多色向量空间拉姆齐数\(R_2(t;k)\)，通过二元单纯码将问题简化为经典超图拉姆齐问题，给出\(R_2(t;k)\)上界，改进了此前结果，还得到二元射影空间色数下界，\(t = 2\)时恢复与三角形多色拉姆齐数联系。
AI 中文摘要
对于每个固定整数\(t\geq2\)，我们给出了多色向量空间拉姆齐数\(R_2(t;k)\)的一个上界，它是一个高度与\(k\)无关的塔函数。对于\(t\geq3\)，这是其形式的首个界，显著改进了早期高度为\(k\)的线性函数的界。我们通过二元单纯码将问题简化为经典超图拉姆齐问题来实现这一点。特别地，我们证明对于\(r = 2^{t - 1}\)和\(s = 2^t - 1\)，\(R_2(t;k) \leq \left\lceil \log R(K_s^{(r)}; k + 1) \right\rceil \leq \mathrm{twr}_{r - 1}(ck\log k)\)，其中\(R(K_{s}^{(r)}; k + 1)\)是\(s\)个顶点的完全\(r\) - 均匀超图的经典\((k + 1)\) - 色拉姆齐数。这种改进也转化为关于\((t - 1)\) - 平坦的二元射影空间色数的改进下界。对于\(t = 2\)，它恢复了与三角形多色拉姆齐数的联系。
英文摘要
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.
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