Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for the formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time computable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In t...

In general terms, Ramsey Theory studies the properties of structures that are preserved under taking partitions, and covers problems in areas including logic, theoretical computer science, number theory, game theory, and beyond. In this paper, we develop the model theoretic machinery to prove Ramsey’s Theorem, and then use more classical combinatorial methods to get a finer resolution on these ...

Extending classical early Ramsey-type results, the structural Ramsey theory originated at the beginning of 70ies, see [11] for references. However the list of Ramsey classes, as the top of the line of Ramsey properties, was somewhat limited. This was also supported by result of Nešetřil [7] that made a connection between Ramsey classes and ultrahomogeneous structures showing particularly that t...

An ordered graph G< is a graph G with vertices ordered by the linear ordering <. The ordered Ramsey number R(G<, c) is the minimum number N such that every ordered complete graph with c-colored edges and at least N vertices contains a monochromatic copy of G<. For unordered graphs it is known that Ramsey numbers of graphs with degrees bounded by a constant are linear with respect to the number ...

Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G, H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120–133] asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved tha...

Given a labeled graph H with vertex set {1, 2, . . . , n}, the ordered Ramsey number r<(H) is the minimum N such that every two-coloring of the edges of the complete graph on {1, 2, . . . , N} contains a copy of H with vertices appearing in the same order as in H. The ordered Ramsey number of a labeled graph H is at least the Ramsey number r(H) and the two coincide for complete graphs. However,...

We conduct a model-theoretic investigation of three infinitary ramseyan statements: Ramsey Theorem for pairs and two colours (RT2), Canonical Ramsey Theorem for pairs (CRT) and Regressive Ramsey Theorem for pairs (RegRT). We prove theorems that approximate the logical strength of these principles by the strength of their finite iterations known as density principles. We then investigate their l...