Group ramsey theory

AbstractA subset S of a group G is said to be a sum-free set if S ∩ (S + S) = ⊘. Such a set is maximal if for every sum-free set T ⊆ G, we have |T| ⩽ |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ⊆ T ⊆ G, we have S = T. Properties of locally maximal sum-free sets are studied and the sets are determined (up to isomorphism) for groups of small order

On the dimension growth of groups

Dimension growth functions of groups have been introduced by Gromov in 1999. We prove that every solvable finitely generated subgroups of the R. Thompson group $F$ has polynomial dimension growth while the group $F$ itself, and some solvable groups of class 3 have exponential dimension growth with exponential control. We describe connections between dimension growth, expansion properties of finite graphs and the Ramsey theory.Comment: 20 pages; v3: Erratum and addendum included as Section 9. We can only prove that the lower bound of the dimension growth of $F$ is exp sqrt(n). New open questions and comments are added. v4: The paper is completely revised. Dimension growth with control is introduced, connections with graph expansion and Ramsey theory are include

Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures

It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original destription by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fra\"iss\'e theory and Ramsey theory. This work generalizes some of the known results about countable structures.Comment: 12 page

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

Universal minimal flows of automorphism groups

We investigate some connections between the Fraïssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As a result we compute the universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ∞-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory: the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa

Tur\'an and Ramsey problems for alternating multilinear maps

Guided by the connections between hypergraphs and exterior algebras, we study Tur\'an and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and has origins in the works of Lov\'asz (Proc. Sixth British Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and Feldman and Propp (Adv. Math., 1992). Our main result is a Ramsey theorem for alternating bilinear maps. Given $s, t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$ subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace $W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural group-theoretic (for finite $p$-groups) and geometric (for Grassmannians) implications, and leads to new Ramsey-type questions for varieties of groups and Grassmannians.Comment: 20 pages. v3: rewrite introductio