向量空间的Ramsey理论参数化 I：离散空间

Big Ramsey degrees in the metric setting

Oscillation stability is an important concept in Banach space theory which happens to be closely connected to discrete Ramsey theory. For example, Gowers proved oscillation stability for the Banach space $c_0$ using his now famous Ramsey theorem for $\mathrm{FIN}_k$ as the key ingredient. We develop the theory behind this connection and introduce the notion of compact big Ramsey degrees, extending the theory of (discrete) big Ramsey degrees. We then prove existence of compact big Ramsey degrees for the Banach space $\ell_\infty$ and the Urysohn sphere, with an explicit characterization in the …

Ramsey theory for hypergroups

In this paper, Ramsey theory for discrete hypergroups is introduced with emphasis on polynomial hypergroups, discrete orbit hypergroups and hypergroup deformations of semigroups. In this context, new notions of Ramsey principle for hypergroups and $α$-Ramsey hypergroup, $0 \leq α<1,$ are defined and studied.

四个 Gallai–Ramsey 型猜想的反证

Disproofs of four Gallai-Ramsey-type conjectures

本文通过构造新的下界反证了关于扇形、轮形和 kipas 的三个 Gallai–Ramsey 数猜想，并给出两类 Ramsey-full 但非 Gallai–Ramsey-full 的图，否定相关等价猜想。

Turán and Ramsey Properties of Subcube Intersection Graphs

The discrete cube $\{0,1\}^d$ is a fundamental combinatorial structure. A subcube of $\{0,1\}^d$ is a subset of $2^k$ of its points formed by fixing $k$ coordinates and allowing the remaining $d-k$ to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no $r+1$ of them have non-empty intersection, how many pairwise intersections can …

素数、阿贝尔群及其他离散结构的典型拉姆齐性质

Typical Ramsey properties of the primes, abelian groups and other discrete structures

本文在阿贝尔群一般框架下研究随机 Rado 问题，提出随机 Rado 引理作为通用工具，并结合超饱和结果确定多种情形的概率阈值，进而得到素数 van der Waerden 定理的随机版本及整数格上的推广。

Large subsets of discrete hypersurfaces in $\mathbb{Z}^d$ contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many colline…