# TLC Spring 2024

**Twenty-fourth meeting: Saturday, March 23, 202**4

** Location: **UNC Greensboro — Sullivan Science Building, SULV 101, and the first-floor lobby.

**Parking:** Parking is free at the McIver Deck. The gate will be down, but pull a ticket to enter. The gates will raise from 5 PM to 6 PM to allow for free exit. If you need to leave at some other time, buzz the intercom at the gate and inform the answering staff member of your attendance at the conference; the staff member would raise the gate remotely.

**Speakers:**

Sean English (UNC Wilmington), Emily Heath (Iowa State University), Jacob Matherne (NCSU), and Cosmin Pohoata (Emory University)

** Tentative Conference Schedule** (all times in EST)

(Abstracts can be found below)

09:00 – 10:00am, welcome

10:00 – 11:00am, **Sean English**

11:00 – 11:30am, coffee break

11:30 – 12:30pm, **Jacob Matherne**

12:30 – 2:30pm, lunch break

02:30 – 3:30pm, **Emily Heath**

03:30 – 4:00pm, coffee break

04:00 – 5:00pm,** Cosmin Pohoata**

** Registration:**To register, please fill out this form. This form includes a section with questions regarding funding.

** Lunch break: **Tate Street has many dining options and is a ten-minute walk from the conference building. If you eat at Boba House, go early as I have always experienced bad wait times, well over 30 minutes.

** Organizing Committee:** Laura Colmenarejo (NCSU), Sean English (UNC Wilmington) and Clifford Smyth (UNC Greensboro)

**Funding provided by NSF**

### Titles and Abstracts

**Sean English:** Rational Exponents for Generalized Turán Numbers

The extremal number of the graph F, denoted *ex(n, F),* is the maximum number of edges in an F-free graph on n vertices. The inverse rational exponent conjecture (perhaps first posed by Erdös and Simonovits in ’81) postulates that for each rational number *r* ∈ [1, 2], there exists some graph F such that *ex(n, F) = Θ(n^r)*.

Recently, Bukh and Conlon proved a slightly weaker version of this conjecture – if one allows for finite families of forbidden graphs, then such a family does exist for each rational *r.*

We will show that a generalization of this conjecture also holds. Given two graphs F and H, the generalized extremal number *ex(n, H, F)* is the maximum number of copies of H in an F-free graph on n vertices (note that *ex(n, F) = ex(n, K2, F)*). We will explore which rational exponents are realizable for some different graphs H.

This is joint work with Anastasia Halfpap and Bob Krueger.

**Emily Heath: **Conflict-free hypergraph matchings and generalized Ramsey numbers

Given graphs $G$ and $H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring in which each copy of $H$ receives at least $q$ colors. Let $f(G,H,q)$ be the minimum number of colors which are required for an $(H,q)$-coloring of $G$. Determining $f(K_n,K_p,2)$ for all $n$ and $p$ is equivalent to determining the classical multicolor Ramsey numbers. Recently, Mubayi and Joos introduced the use of a new method for proving upper bounds on these generalized Ramsey numbers by finding a “conflict-free” matching in an appropriate auxiliary hypergraph. In this talk, we will show how to generalize their approach to give bounds on the generalized Ramsey numbers for several families of graphs. This is joint work with Deepak Bal, Patrick Bennett, and Shira Zerbib.

**Jacob Matherne**: Examples of Lorentzian polynomials

Lorentzian polynomials were introduced by Brändén and Huh in 2019. They are a generalization of homogeneous stable polynomials, and they have recently been used to prove log-concavity properties for a number of interesting polynomials in a variety of areas of mathematics.

The purpose of this talk is to share some families of polynomials that are either known or expected to be Lorentzian, especially from the areas of algebraic combinatorics and representation theory. Two important classes I will focus on are the (normalizations of) Schur polynomials and the chromatic symmetric functions of certain graphs appearing in the Stanley-Stembridge conjecture. The talk will be based on joint work with June Huh, Karola Mészáros, Alejandro Morales, Jesse Selover, and Avery St. Dizier.

**Cosmin Pohoata: **The Heilbronn triangle problem

Given an integer $n \geq 3$, the Heilbronn triangle problem asks for the smallest number $\Delta = \Delta(n)$ such that in every configuration of $n$ points in the unit square $[0,1]^2$ one can always find three among them which form a triangle of area at most $\Delta$. A trivial upper bound of the form $\Delta = O(1/n)$ follows from the simple observation that if one partitions $[0,1]^2$ into $n/2$ vertical strips of width $2/n$, then one of these strips must contain at least $3$ of the points. The problem of improving upon this basic observation is a difficult one, with a long and rich history. We will survey some of this history and discuss with some new connections between this problem and various themes in incidence geometry and projection theory. Based on joint work with Alex Cohen and Dmitrii Zakharov.