Aart Blokhuis: Flat-containing and shift-blocking sets
For a pair of non-negative integers $r\ge d$, how small can a
subset $G\subset V= V(r,2)=GF(2)^r$ be, given
that for any vector $v\in V$, there is a $d$-flat passing through $v$ and
contained in $G\cup\{v\}$?
Equivalently, but looking at the complement:
how large can a subset $B\subset V$ be, given
that for any $v\in V$ there is a linear $d$-subspace not blocked
non-trivially by the translate $B+v$? We prove a number of lower
and upper bounds.