Any valid Ramsey statement n −→ (k)2 can be encoded into a DNF formula RAM(n, k) of size O(nk) and with terms of size (k 2 ) . Let rk be the minimal n for which the statement holds. We prove that RAM(rk, k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that ther...

We define a mild generalization of classical Fraïssé classes, and in this context, we recover several old (often with novel proofs) and new facts about Ramsey classes and generalized indiscernibles in the sense of [12], [17], and other sources. Of particular novelty are: (1) a new and clearly model-theoretic proof of the equivalence of the Ramsey property and the extreme amenability of the auto...

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an nO(log(n)) upper bound on the Ramsey number of a path wit...

In this paper we exhibit a similarity between Euclidean Ramsey problems and Knaster-type problems. By borrowing ideas from Ramsey theory we prove weak Knaster properties of non-equatorial triangles in spheres, and of simplices in Euclidean spaces.

We compare tools for complementing nondeterministic Büchi automata with a recent termination-analysis algorithm. Complementation of Büchi automata is a key step in program verification. Early constructions using a Ramsey-based argument have been supplanted by rank-based constructions with exponentially better bounds. In 2001 Lee et al. presented the size-change termination (SCT) problem, along ...

We present one novel result and two novel proofs of previously known results in structural Ramsey theory. Regarding the former, we give a new characterization of the Ramsey property for a Fraïssé class in terms colorings of induced substructures of its generic model (or Fraïssé limit). This result is obtained as a corollary of a more general theorem in which we show that under relatively mild h...

Given a graph H and an integer r ≥ 2, let G → (H, r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = maxF⊆G |E(F )|/|V (F )| and define the Ramsey density minf (H, r) as the infimum of m(G) over all graphs G such that G → (H, r). In the first part of this paper we show that when H is a complete graph ...

Ramsey theory is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a specific size. This paper will explore some basic definitions of and history behind Ramsey theory, but will focus on a subsection of Ramsey theory known as Ramsey numbers. A discussion of what Ramsey numbers are, some examples of their relevance in real-life scenarios, and a...

In 2012 M. Sokić proved that the that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. Böttcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Sokić. Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permu...