Polynomial Schur's theorem

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$.Comment: 21 page

Enumeration of three term arithmetic progressions in fixed density sets

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3: Incorporated feedbac

Mini-Workshop: Hypergraph Turan Problem

This mini-workshop focused on the hypergraph Turán problem. The interest in this difficult and old area was recently re-invigorated by many important developments such as the hypergraph regularity lemmas, flag algebras, and stability. The purpose of this meeting was to bring together experts in this field as well as promising young mathematicians to share expertise and initiate new collaborative projects

An exploration of anti-van der Waerden numbers

In this paper results of the anti-van der Waerden number of various mathematical objects are discussed. The anti-van der Waerden number of a mathematical object G, denoted by aw(G,k), is the smallest r such that every exact r-coloring of G contains a rainbow k-term arithmetic progression. In this paper, results on the anti-van der Waerden number of the integers, groups such as the integers modulo n, and graphs are given. A connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established. The anti-van der Waerden number of [m]X[n] is explored. Finally, connections between anti-van der Waerden numbers, rainbow numbers, and anti-Schur numbers are discussed

On the complexity of finding and counting solution-free sets of integers

Given a linear equation $\mathcal{L}$, a set $A$ of integers is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving $\mathcal{L}$-free sets of integers. The main questions we consider involve deciding whether a finite set of integers $A$ has an $\mathcal{L}$-free subset of a given size, and counting all such $\mathcal{L}$-free subsets. We also raise a number of open problems.Comment: 27 page