# Triangle Lectures in Combinatorics

These are a series of combinatorial workshops, held once per semester, each on a Saturday. The series began with its first meeting in Spring 2010. They rotate among the universities in and around the Research Triangle. Participants come from numerous colleges and universities within a few hours drive, and some from even farther away. These workshops are funded by the National Science Foundation, in particular enabling us to bring in four exciting speakers to give one hour talks each time as well as funding travel expenses for participants.

**TLC steering committee: **Laura Colmenarejo (NCSU), Sean English (UNC Wilmington), Gabor Pataki (UNC Chapel Hill), Clifford Smyth (UNC Greensboro), and Fan Wei (Duke University)

**Former ****TLC steering committee:** Nathan Reading (NCSU), Ricky Liu (NCSU), Ezra Miller (Duke University), and Gabor Pataki (UNC Chapel Hill).

# 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.

**Past meetings:**

**Twenty third meeting:** Saturday, November 18, 2023 at NCSU **Speakers:** David Galvin (University of Notre Dame, US), Pamela E. Harris (University of Wisconsin at Milwaukee, US), Jianping Pan (North Carolina State University, US), and Fan Wei (Duke University, US)**Organizing Committee: **Laura Colmenarejo (NCSU), Clifford Smyth (UNC Greensboro) **More information**

**Twenty second meeting:** Saturday, April 8, 2023 at NCSU **Speakers:** Ed Allen (Wake Forest University, US), Nantel Bergeron (York University, Canada), Carly Klivans (Brown University, US), and Bruce Sagan (Michigan State University, US)**Organizing Committee: **Laura Colmenarejo (NCSU), Sarah K. Mason (Wake Forest University), Nick Mayers (NCSU), Clifford Smyth (UNC Greensboro) **More information**

**Twenty first meeting:** Saturday, December 3, 2022 at UNC Greensboro (in person)**Speakers:** Jeremy L. Martin (University of Kansas), Erik Slivken (University of North Carolina Wilmington), Catherine Yan (Texas A&M University), and Yan Zhuang (Davidson College)**Organizing Committee: **Laura Colmenarejo (NCSU), Sarah K. Mason (Wake Forest University), Clifford Smyth (UNC Greensboro) **More information**

**Twentieth meeting:** Saturday, November 14, 2020 at UNC Greensboro (online)**Speakers:** Sara Billey (University of Washington, Seattle), Hao Huang (Emory University), Igor Pak (University of California, Los Angeles), and Nathan Williams (University of Texas, Dallas)**Organizing Committee: **Clifford Smyth (UNC Greensboro), Thomas McConville (Kennesaw State U.), Cynthia Vinzant (NCSU)**More information**

**Nineteenth meeting:** Saturday, February 29, 2020 at UNC Charlotte

**Speakers: **Margaret Bayer (U Kansas), Michel Goemans (MIT), Svetlana Poznanović (Clemson), Cynthia Vinzant (NCSU)

**Organizing Committee: **Gabor Hetyei (UNC Charlotte), Ricky Liu (NCSU), Gabor Pataki (UNC Chapel Hill)

**More information**

**Eighteenth meeting:** Saturday, November 16, 2019** **at NCSU

**Speakers: **Georgia Benkart (University of Wisconsin at Madison), Javier Peña (Carnegie Mellon University), Margaret Readdy (University of Kentucky), Alexander Yong (University of Illinois at Urbana-Champaign)

**Organizing Committee: **Patricia Hersh (NCSU), Gabor Hetyei (UNC-Charlotte), Kailash Misra (NCSU), Gabor Pataki (UNC Chapel Hill)

**More information**

**Seventeenth meeting:** Saturday, March 30, 2019 at** **Wake Forest University **Speakers:** James Haglund (University of Pennsylvania), Adam Marcus (Princeton University), Jennifer Morse (University of Virginia), Nathan Reading (North Carolina State University)**Organizing committee: **Sarah Mason (Wake Forest), Ed Allen (Wake Forest), Patricia Hersh (NCSU), Cliff Smyth (UNC Greensboro)**More information**

**Sixteenth meeting:** November 10, 2018 at UNC Chapel Hill**Speakers:** László Babai (University of Chicago), Daniel Bienstock (Columbia), Rebecca Goldin (George Mason University), and Lek-Heng Lim (University of Chicago).**Organizing Committee:** Prakash Belkale (UNC Chapel Hill), Gabor Pataki (UNC Chapel Hill), Richard Rimanyi (UNC Chapel Hill), and Seth Sullivant (NCSU).**More information**

**Fifteenth meeting:** March 26, 2018 at NCSU**Speakers:** Richard Kenyon (Brown), Karola Mészáros (Cornell), Gabor Hetyei (UNC-Charlotte), and Seth Sullivant (NCSU).**Organizing Committee:** Patricia Hersh (NCSU), Ricky Liu (NCSU), and Cynthia Vinzant (NCSU)**More information**

**Fourteenth meeting:** November 19, 2016 at NCSU**Speakers:** Bruno Benedetti (U. Miami), Maria Chudnovsky (Princeton), Jeffery Lagarias (U. Michigan), Josephine Yu (Georgia Tech).**Organizing Committee:** Ricky Liu (NCSU), Seth Sullivant (NCSU), and Cynthia Vinzant (NCSU)**More information**

**Thirteenth meeting:** Saturday February 27, 2016 at UNC Greensboro**Speakers:** Penny Haxell (U. Waterloo), Jeff Kahn (Rutgers), Greta Panova (U. Penn), Peter Winkler (Dartmouth).**Organizing Committee:** Clifford Smyth, chair (UNC Greensboro), David Galvin (Notre Dame), and Patricia Hersh (NCSU).**More information**

**Twelfth meeting:** Saturday October 3, 2015 at Duke**Speakers:** Yuliy Baryshnikov (UIUC), Susan Holmes (Stanford), Dana Randall (Georgia Tech), and John Shareshian (Washington University in St. Louis).**Organizing Committee:** Sayan Mukherjee, chair (Duke), John Harer (Duke), Patricia Hersh (NCSU), and Mauro Maggioni (Duke)**More information**

**Eleventh meeting:** February 21, 2015 at NCSU**Speakers: **Matthew Baker (Georgia Tech), Henry Cohn (Microsoft New England), Lionel Levine (Cornell), and Anne Schilling (UC Davis).**Organizing Committee:** Ricky Liu (NCSU), Seth Sullivant (NCSU), and Cynthia Vinzant (NCSU)**More information**

**Tenth meeting:** October 4, 2014, at High Point University**Speakers: **Richard Ehrenborg (Kentucky), Robert Ghrist (Penn), Luis Serrano (U Quebec a Montreal), and Rekha Thomas (U. Washington).**Organizing Committee:** Lindsay Piechnik (chair, High Point), Ed Allen (Wake Forest), Sarah Mason (Wake Forest), Seth Sullivant (NCSU), and Laurie Zack (High Point)**More information**

**Ninth meeting:** February 22, 2014 at UNC-Chapel Hill.**Speakers:** Shankar Bhamidi (UNC-Chapel Hill), Anders Buch (Rutgers), Pablo Parrilo (MIT), Eva Tardos (Cornell)**Organizing committee: **Gabor Pataki (UNC-Chapel Hill), Lindsay Piechnik (High Point University), Scott Provan (UNC-Chapel Hill), Richard Rimanyi (UNC-Chapel Hill), Jack Snoeyink (UNC-Chapel Hill)**More information**

**Eighth meeting:** September 21, 2013 at NCSU.**Speakers: **George Andrews (Penn State), Matt Beck (San Francisco State University), Robin Pemantle (University of Pennsylvania), and Victoria Powers (Emory University)**Organizing committee: **Michael Singer (NCSU), Carla Savage (NCSU), and Seth Sullivant (NCSU)**More information**

**Seventh meeting:** February 9, 2013 at Wake Forest University.**Speakers:** Louis Billera (Cornell), Rod Canfield (University of Georgia), Matthew Kahle (Ohio State University), Michelle Wachs (University of Miami)**Organizing committee:** Sarah Mason (chair, Wake Forest University), Ed Allen (Wake Forest University), Alex Fink, (NCSU), Patricia Hersh (NCSU)**More information**

**Sixth meeting:** September 22, 2012 at NCSU.**Speakers:** Allen Knutson (Cornell), Vin de Silva (Pomona College), Richard Stanley (MIT), Lauren Williams (UC Berkeley).**Organizing committee:** Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).**More information**

**Fifth meeting:** February 11, 2012 at Duke University.**Speakers:** Alex Fink (NCSU), Sergey Fomin (Michigan), Nets Katz (Indiana University), Isabella Novik (University of Washington).**Organizing Committee:** Christine Berkesch (Duke), Sonja Mapes (Duke), Ezra Miller (Duke).**More information**

**Fourth meeting:** November 5, 2011 at UNC Chapel Hill.**Speakers:** Thomas Lam (Michigan), Jesus De Loera (UC Davis), Ezra Miller (Duke), Doron Zeilberger (Rutgers)**Organizing committee:** Prakash Belkale (UNC Chapel Hill), Gabor Pataki (UNC Chapel Hill), Robert Proctor (UNC Chapel Hill), Scott Provan (UNC Chapel Hill), Richard Rimanyi(UNC Chapel Hill).**More information**

**Third meeting:** April 9, 2011 at NCSU.**Speakers:** Prakash Belkale (UNC Chapel Hill), Vic Reiner (University of Minnesota), John Stembridge (University of Michigan), Stephanie van Willigenburg (UBC).**Organizing committee:** Hoda Bidkhori (NCSU), Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).**More information**

**Second meeting:** September 25, 2010 at Duke.**Speakers:** Alexander Barvinok (University of Michigan), Anne Shiu (Duke), Sami Assaf (MIT), Persi Diaconis (Stanford).**Organizing committee:** Patricia Hersh (NCSU), Sonja Mapes (Duke), Ezra Miller (Duke).**More information**

**First meeting:** February 6, 2010 at NCSU.**Speakers:** Carla Savage (NCSU), Bernd Sturmfels (UC Berkeley), Ed Swartz (Cornell), Laszlo Szekely (University of South Carolina).**Organizing committee:** Patricia Hersh (NCSU), Ezra Miller (Duke), Scott Provan (UNC), Nathan Reading (NCSU).**More information**