Speaker: Brahadeesh Sankarnarayanan
Day/Date/Time: 29th May, 2024, Wednesday, 11:30 AM
Venue: Ramanujan Hall
Title: Sunflowers, symmetric designs and tournaments
Abstract: For a fraction a/b in (0,1), a family F of subsets of [n] := {1,
..., n} is called a "fractional (a/b)-intersecting family" if, for every
pair of distinct sets A, B in F, we have |A \cap B| = a/b |A| or a/b |B|.
The natural extremal question is: How large can an a/b-intersecting family
over [n] be? This notion was introduced in Balachandran邦athew邦ishra
(Electron. J. Combin. 26 (2019), #P2.40), wherein they showed that |F|<
O(n log n), and they gave constructions of a/b-intersecting families of
size at least O(n). The conjecture (which is still open) is that |F|<O(n)
for any a/b-intersecting family F over [n]. In this talk, I will discuss
some recent progress on this conjecture, and some related questions
concerning ranks of certain matrix ensembles, tournaments, symmetric
designs, and sunflowers.