论文标题
通过多项式理想的镜头和无效的Ramsey数字
Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze
论文作者
论文摘要
在本文中，我们通过Hilbert的Nullstellensatz和Alon的Combinatorial Nullstellensatz研究了Ramsey Number Number $ R（R，S）$。我们提供了多项式编码，其解决方案对应于订单$ n $的Ramsey图，那些不包含$ k_r $或$ \ bar {k} _s $的副本。当这些系统没有解决方案和$ n \ ge r（r，s）$时，我们构建了Nullstellensatz证书，其学位等于Conlon，Fox，Grinshpun和He引入的受限制的在线Ramsey号码。此外，我们表明，这些结果在拉姆西理论中推广到其他数字，包括Rado，Van der Waerden和Hales-Jewett数字。最后，我们介绍了一个与某个“拉姆西多项式”系数有关的数字系列，该系数为拉姆西数字提供了下限。
In this article we study the Ramsey numbers $R(r,s)$ through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order $n$, those that do not contain a copy of $K_r$ or $\bar{K}_s$. When these systems have no solution and $n \ge R(r,s)$, we construct Nullstellensatz certificates whose degrees are equal to the restricted online Ramsey numbers introduced by Conlon, Fox, Grinshpun and He. Moreover, we show that these results generalize to other numbers in Ramsey theory, including Rado, van der Waerden, and Hales-Jewett numbers. Finally, we introduce a family of numbers that relate to the coefficients of a certain "Ramsey polynomial" that gives lower bounds for Ramsey numbers.
