## Presentation Type

Oral Presentation

## Category

STEM (science, technology, engineering, mathematics)

## Abstract/Artist Statement

Suppose everyone at Gradcon listed their top 5 favorite movies of all time. To save everyone the trouble of ranking, we'll assume the lists are unordered. Here's the question: how many lists do we need to guarantee we have three lists, any two of which share the exact same titles? For example, three lists that share nothing in common would satisfy our condition. It turns out that such an event is guaranteed if we have 3840 lists. This seems rather large, and is probably far too many, but trying to find a better upper bound has been a project of extremal set theory for over fifty years!

To generalize this problem, we consider sets (unordered collections of objects) and families, which are sets of sets. Extremal set theory is concerned with how large (or small) a family of sets can be if they must satisfy certain restrictions. A relatively young discipline of mathematics, its earliest problems continue to remain unsolved to this day. The sunflower conjecture is one such problem.

A family of sets is called a sunflower if each possible pairwise intersection (the set induced by including only objects included in a pair of sets) produces the same subset. Given the size of the sets in a family, the sunflower lemma establishes sufficient conditions for finding a sunflower subfamily of any predetermined size. Unfortunately, the bound determined by the lemma is quite large. Without getting too technical, the sunflower conjecture posits that the true upper bound is, for the most part, much smaller. As the existence of a sunflower can imply further structural properties, such a result would have far-reaching effects.

This presentation serves as an introduction to the sunflower lemma (and its associated conjecture) for a general audience. A willingness to learn is the only prerequisite.

## Mentor Name

Cory Palmer

## Personal Statement

As someone who values the discussion of ideas, but is also a mathematician, I understand that what I happen to think about daily is not so easily communicable. For this reason, my goal for this talk (and generally most talks) is to spark curiosity. To practice mathematics is to be perpetually confused, a feeling unattainable without a healthy amount of curiosity. So, though the ideas discussed might appear out of reach, it is that feeling of being utterly lost which pushes me (and I suspect others) to figure out (perhaps only a smidge of) what's going on. Extremal set theory is a rich and developing discipline of mathematics which I'm thankful to be a part of. While my own research makes use of structures similar to sunflowers (which we refer to simply as flowers), I believe the best way to encourage engagement with higher level mathematics is to talk about the big problems that propel research. Otherwise, it's only the select few of us who have any idea what we're doing. And while I'm glad I understand what I'm saying, it's an even better feeling for someone else to as well.

*Wood oral presentation video*

The Sunflower Lemma

UC 333

Suppose everyone at Gradcon listed their top 5 favorite movies of all time. To save everyone the trouble of ranking, we'll assume the lists are unordered. Here's the question: how many lists do we need to guarantee we have three lists, any two of which share the exact same titles? For example, three lists that share nothing in common would satisfy our condition. It turns out that such an event is guaranteed if we have 3840 lists. This seems rather large, and is probably far too many, but trying to find a better upper bound has been a project of extremal set theory for over fifty years!

To generalize this problem, we consider sets (unordered collections of objects) and families, which are sets of sets. Extremal set theory is concerned with how large (or small) a family of sets can be if they must satisfy certain restrictions. A relatively young discipline of mathematics, its earliest problems continue to remain unsolved to this day. The sunflower conjecture is one such problem.

A family of sets is called a sunflower if each possible pairwise intersection (the set induced by including only objects included in a pair of sets) produces the same subset. Given the size of the sets in a family, the sunflower lemma establishes sufficient conditions for finding a sunflower subfamily of any predetermined size. Unfortunately, the bound determined by the lemma is quite large. Without getting too technical, the sunflower conjecture posits that the true upper bound is, for the most part, much smaller. As the existence of a sunflower can imply further structural properties, such a result would have far-reaching effects.

This presentation serves as an introduction to the sunflower lemma (and its associated conjecture) for a general audience. A willingness to learn is the only prerequisite.