Related papers: Big Ramsey degrees using parameter spaces

The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be…

We prove the Erd\H os--S\'os conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a…

We use $p$-rank bounds on partial ovoids and the classical bounds on Ramsey numbers to obtain upper bounds on the size of partial $m$-ovoids in finite classical polar spaces. These bounds imply non-existence of $m$-ovoids for new infinite…

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and…

We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the…

We study graphs with nonnegative Bakry-\'Emery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is…

We study an inhomogeneous sparse random graph on [N] = {1, . . . , N } as introduced in a seminal paper by Bollobas, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices…

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…

Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs…

We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. We show…

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain…

We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and…

We study the global structure of Lorentzian manifolds with partial sectional curvature bounds. In particular, we prove completeness theorems for homogeneous and isotropic cosmologies as well as static spherically symmetric spacetimes. The…

Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open…

We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…

We prove an infinite-dimensional version of an approximate Ramsey theorem of Gowers, initially used to show that every Lipschitz function on the unit sphere of $c_0$ is oscillation stable. To do so, we use the theory of ultra-Ramsey spaces…

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique…

We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a…

We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete…

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter…