Related papers: Open questions about Ramsey-type statements in rev…
Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it…
We consider finitary approximations of the (embedding) Ramsey property. Using a class of homogeneous reducts of random ordered hypergraphs, we prove that these properties form a strict hierarchy. We also show that every class of finite…
We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF.…
A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some…
We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a…
We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion…
A function is diagonally non-computable (d.n.c.) if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which…
Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the…
An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting…
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…
We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided…
Ramsey's Theorem states that a graph $G$ has bounded order if and only if $G$ contains no complete graph $K_n$ or empty graph $E_n$ as its induced subgraph. The Gy\'arf\'as-Sumner conjecture says that a graph $G$ has bounded chromatic…
Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in…
For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show…
We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a…
We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$…
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…
If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently…
Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function…
Let $\mathsf{TT}^2_k$ denote the combinatorial principle stating that every $k$-coloring of pairs of compatible nodes in the full binary tree has a homogeneous solution, i.e. an isomorphic subtree in which all pairs of compatible nodes have…