The no-three-in-line problem on a torus Joint work with Andrew Groot, Deven Pandya, Bart Snapp

For a group , let denote the cardinality of the largest subset so that no three elements of are in the same coset of a cyclic subgroup. Undergraduates Andrew Groot and Deven Pandya, advised by myself and my colleague Bart Snapp, considered the case , and showed that This problem can also be formulated as a Gröbner basis question; after doing so, we computed for and .

Thinking of a coset of a cyclic subgroup as a “line,” there is then connection with the usual “no three in line problem.” Paul Erdős proved that for a prime , one can place points on the lattice in the plane [1]; the construction goes via a parabola modulo . Other more complicated constructions manage to place more points [2].

My interest lately has been considering the question for other groups. Although the no-three-in-line problem for can be considered as the -arc problem from projective geometry [3], the question is interesting for, say, or where, say, Bezout’s theorem doesn’t make sense anymore.