Line-free sets in Fnp

For primes p we study subsets of Fnp that do not contain a full line and try to maximize the cardinality of such subsets.

It is clear that a cube of sidelength p−1 will not contain a full line and in 1977 Jamison showed that this construction is optimal in the plane. However, not much was known for higher dimension.

A blogpost of Douglas Zare in 2016 shows that the cube construction, already in three dimensions, is not optimal, but only by improving the lower bound by 1.

Our work is the first attempt to significantly improve this construction. In this presentation we will present new upper and lower bounds for the three-dimensional case and their consequences to higher dimensions, aswell as higher dimensional generalizations.

This is joint work with Christian Elsholtz, Erik Füredi, Benedek Kovács, Péter Pál Pach, Dániel Simon and Nóra Velich.