Maker-Breaker Rado games for equations with radicals

We study two-player positional games where Maker and Breaker take turns to select a previously unoccupied number in $\{1,2,\ldots,n\}$. Maker wins if the numbers selected by Maker contain a solution to the equation $$x_1^{1/\ell}+\cdots+x_k^{1/\ell}=y^{1/\ell}$$ where $k$ and $\ell$ are integers with $k\geq2$ and $\ell\neq0$, and Breaker wins if they can stop Maker. Let $f(k,\ell)$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\ldots,x_k$ are not necessarily distinct, and let $f^*(k,\ell)$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\ldots,x_k$ are distinct. When $\ell\geq1$, we prove that, for all $k\geq2$, $f(k,\ell)=(k+2)^\ell$ and $f^*(k,\ell)=(k^2+3)^\ell$; when $\ell\leq-1$, we prove that $f(k,\ell)=[k+\Theta(1)]^{-\ell}$ and $f^*(k,\ell)=[\exp(O(k\log k))]^{-\ell}$. Our proofs use elementary combinatorial arguments as well as results from number theory and arithmetic Ramsey theory.Comment: 18 pages, 1 figur

Independence and counting problems in Combinatorics and Number theory

The method ofhypergraph containers has become a very important tool for dealing with problems which can be phrased in the language of independent sets in hypergraphs. This method has applications to numerous problems in combinatorics and other areas. In this thesis we consider examples of such problems; in particular problems concerning sets avoiding solutions to a given system of linear equations L (known as L-free sets) or graphs avoiding copies of a given graph H (H-free graphs). First we attack a number of questions relating to L-free sets. For example, we give various bounds on the number of maximal L-free subsets of [n] for three-variable homogeneous linear equations L. We then use containers to prove results corresponding to problems concerning tuples of disjoint independentsets in hypergraphs. In particular we generalise the random Ramsey theorem of Rodl and Rucinski by providing a resilience analogue. We obtain similar results for L-free sets. Finally we consider the Maker-Breaker game where Maker's aim is to obtain a solution to a given system of linear equations L amongst a random set of integers. We determine the threshold probability for this game for a large class of systems L