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 $\{1,2,\ldots,n\}$. 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 $f(k,\ell)=[k+\Theta_k(1)]^{-\ell}$ and $f^*(k,\ell)=[\exp(O_k(k\log k))]^{-\ell}$. Our proofs use elementary combinatorial arguments as well as results from number theory and arithmetic Ramsey theory.