Maker-Breaker Rado games for equations with radicals

Abstract

We study two-player positional games where Maker and Breaker take turns to select a previously unoccupied number in $\{1,2,\dots ,n\}$. 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 $k\ge 2$ and $\ell \ne 0$, 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 $x_1,\dots ,x_k$ are not necessarily distinct, and let $f^{\ast }(k,\ell )$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\dots ,x_k$ are distinct. When $\ell \ge 1$, we prove that, for all $k\ge 2$, $f(k,\ell )=(k+2{)}^{\ell }$ and $f^{\ast }(k,\ell )=(k^2+3{)}^{\ell }$; when $\ell \le -1$, we prove that $f(k,\ell )=[k+{\Theta }_k(1){]}^{-\ell }$ and $f^{\ast }(k,\ell )=[\exp (O_k(k\log k)){]}^{-\ell }$. Our proofs use elementary combinatorial arguments as well as results from number theory and arithmetic Ramsey theory.

arXiv comment18 pages, 1 figure