We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller’s conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7=128$. We present an automated method to ...${\mathbb{}}^{\mathrm{}}$${\mathbb{}}^{\mathrm{}}$

Let S be a set of arbitrary objects, and let $s\mapsto s^{\prime }$ be a permutation of S such that $s^{\prime \prime }={(s^{\prime })}^{\prime }=s$ and $s^{\prime }\ne s$. Let $S^d=\{v_1\cdots v_d:v_i\in S\}$. Two words $v,w\in S^d$ are dichotomous if $v_i=w_i^{\prime }$ for some $i\in [d]=\{1,\cdots ,d\}$, and they form a twin pair if $v_i^{\prime }=w_i$ and $v_j=w_j$ for every ...$$${}^{}$${\mathrm{}}^{}$${\mathbb{}}^{}$${\mathbb{}}^{}$${}_{}{}_{}^{}$${}_{}{}_{}{}_{}$$$${}_{}$$$${}_{}$${}_{}{}_{}^{}$${}_{}{}_{}{}_{}$$$${}^{}$

A family of translates of the unit cube [0,1)^d+T={[0,1)^d+t:t@?T}, T@?R^d, is called a cube tiling of R^d if cubes from this family are pairwise disjoint and @?"t"@?"T[0,1)^d+t=R^d. A non-empty set B=B"1x...xB"d@?R^d is a block if there is a family of ...

Let m1,ź,md be positive integers, and let G be a subgroup of źd such that m1ź ź mdź⊆G. It is easily seen that if a unit cube tiling [0,1)d+t,tźT, of źd is invariant under the action of G, then for every tźT, the number |Tź(t+źd)ź[0,m1) ź [0,md)| is ...

A non-empty subset A of X=X1 źźź Xd is a (proper) box if A=A1 źźź Ad and AiźXi for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: Ai=Bi, Ai=XiźBi, Aiź{Bi,XiźBi}. Let F and G be two ...