Ramsey properties of algebraic graphs and hypergraphs

One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erd\H{o}s from 1947, the random graph on $N$ vertices with edge probability $1/2$, contains no clique or independent set larger than $2\log_2 N$, with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an $r$-uniform hypergraph $\mathcal{H}$ is \emph{algebraic of complexity $(n,d,m)$} if the vertices of $\mathcal{H}$ are elements of $\mathbb{F}^{n}$ for some field $\mathbb{F}$, and there exist $m$ polynomials $f_1,\dots,f_m:(\mathbb{F}^{n})^{r}\rightarrow \mathbb{F}$ of degree at most $d$ such that the edges of $\mathcal{H}$ are determined by the zero-patterns of $f_1,\dots,f_m$. The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity $(n,d,m)$ has good Ramsey properties, then at least one of the parameters $n,d,m$ must be large. In 2001, R\'onyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if $G$ is an algebraic graph of complexity $(n,d,m)$ on $N$ vertices, then either $G$ or its complement contains a complete balanced bipartite graph of size $\Omega_{n,d,m}(N^{1/(n+1)})$. We extend this result by showing that such $G$ contains either a clique or an independent set of size $N^{\Omega(1/ndm)}$ and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for $r$-uniform algebraic hypergraphs that are defined by a single polynomial, that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.Comment: 23 page

Ramsey expansions of metrically homogeneous graphs

We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and the coherent EPPA. Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a "completion algorithm" then allows us to apply several strong results in the areas that imply EPPA and respectively the Ramsey property. The main results have numerous corollaries on the automorphism groups of the Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor revisio

On the Ramsey-Tur\'an number with small ss-independence number

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Tur\'an number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $\alpha_s(G) < f$, where $\alpha_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Tur\'an number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^{\delta})$ for fixed $\delta<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<\delta< 1$, $RT_s(n,K_{s+1}, n^{\delta}) = \Omega(n^{1+\delta-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta}) = O(n^{1+\delta})$. Furthermore, the range of $\delta$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^{\delta})$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.Comment: 25 p