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We consider the Maker-Breaker positional game on the vertices of the $n$-dimensional hypercube ${0,1}^n$ with $k$-dimensional subcubes as winning sets. We describe a pairing strategy which allows Breaker to win when $k = n/4 +1$ in the case where $n$ is a power of 4. Our results also imply the general result that there is a Breakers win pairing strategy for any $n geq 3$ if $k = leftlfloorfrac{3}{7}nrightrfloor +1$.
We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Makers aim being to claim all
The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight
In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio
We determine the asymptotics of the number of independent sets of size $lfloor beta 2^{d-1} rfloor$ in the discrete hypercube $Q_d = {0,1}^d$ for any fixed $beta in [0,1]$ as $d to infty$, extending a result of Galvin for $beta in [1-1/sqrt{2},1]$. M
The main result of the present paper is a new lower bound on the number of Boolean bent functions. This bound is based on a modification of the Maiorana--McFarland family of bent functions and recent progress in the estimation of the number of transv