Title:  Oddtowns, partial ovoids, and the rank-Ramsey problem

Speaker: Prof. Anurag Bishnoi, TU Delft, Netherlands

[https://anuragbishnoi.wordpress.com/]

Abstract :

What is the largest size of a family of subsets of {1, ..., m} such that
every set in the family has odd cardinality and among any three distinct
sets there is at least one pair that intersects in an even number of
elements?  We'll show that this generalisation of the classic Oddtown
problem is equivalent to finding the largest size of a partial 2-ovoid in a
binary symplectic space and that of finding the largest set of nearly
orthogonal vectors in F_2^m. Moreover, we show that it's intimately linked
to the following recently introduced rank-Ramsey problem: for a given m,
find the largest n for which there exists a triangle-free graph with n
vertices whose adjacency matrix A  satisfies rank(A+I) <= m.  If we
consider the rank over the binary field, then the problem is equivalent to
generalized oddtowns, while over other fields it has a different behaviour.

We give new constructions of these objects, improving the state of art,
using a triangle-free Cayley graph associated with BCH codes.  Moreover, by
using binary projective caps, that is, sum-free sets in F_2^n,  we improve
the best construction for this rank-Ramsey problem over the reals.

Joint work with John Bamberg and Ferdinand Ihringer.