## Fall 2022

### Sept 1, 4pm CT

Altgeld Hall (room number 245)

**Speaker:** Igor Pak, UCLA

**Title:** Combinatorial inequalities

**Abstract:** In the ocean of combinatorial inequalities, two islands are especially difficult. First, Mason’s conjectures say that the number of forests in a graph with k edges is log-concave. More generally, the number of independent sets of size k in a matroid is log-concave. These results were established just recently, in a remarkable series of papers by Huh and others, inspired by algebro-geometric considerations.

Second, Stanley’s inequality for the numbers of linear extensions of a poset with value k at a given poset element, is log-concave. This was originally conjectured by Chung, Fishburn and Graham, and famously proved by Stanley in 1981 using the Alexandrov–Fenchel inequalities in convex geometry. No direct combinatorial proof for either result is known. Why not?

In the first part of the talk we will survey these and other combinatorial inequalities. We then mention what does it mean not to have a combinatorial interpretation. Finally, I will briefly discuss our new framework of combinatorial atlas which allows one to give elementary proofs of the two results above, and extend them in several directions. The talk is aimed at the general audience.

### Sept 8, 4pm CT

Altgeld Hall (room number 245)

**Speaker:** Xuding Zhu, Zhejiang Normal University, China

**Title:** List version of the 1-2-3 Conjecture

**Abstract:** The well-known 1-2-3 Conjecture by Karo\'{n}ski, {\L}uczak, and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1, 2 or 3 so that for each edge $uv$ the sums of the weights at $u$ and at $v$ are distinct.

The list version of the 1-2-3 Conjecture by Bartnicki, Grytczuk, and Niwczyk states that the same holds more generally if each edge $e$ has the choice of weights not necessarily from $\{1,2,3\}$, but from any set $\{x(e),y(e),z(e)\}$ of three real numbers.

The goal of this talk is to survey developments on the 1-2-3 Conjecture, especially on the list version of the 1-2-3 Conjecture.

## Spring 2022

### May 18, 11am CT

Altgeld Hall (room number TBD)

**Speaker:** Wojciech Samotij, Tel Aviv University

**Title:** The upper tail for triangles in random graphs

**Abstract:** Let $X$ denote the number of triangles in the random graph $G_{n,p}$. The problem of determining the asymptotics of the logarithimic upper tail probability of $X$, that is, $\log \Pr(X > (1+\delta)\mathbb{E}[X])$, for every fixed positive $\delta$ has attracted considerable attention of both the combinatorics and the probability communities. We shall present an elementary solution to this problem, obtained recently in a joint work with Matan Harel and Frank Mousset. The crux of our approach is a simple probabilistic argument, inspired by the work of Janson, Oleszkiewicz and Ruci\’nski, that reduces the estimation of this upper tail probability to a counting problem

### April 14, 10am CT

245 Altgeld Hall

**Speaker:** Andrew Treglown, U. Birmingham, UK, Fulkerson Prize recipient (2021)

**Title:** Tiling in graphcs

**Abstract:** Given two graphs H and G, an H-tiling in G is a collection of vertex disjoint copies of H in G. Thus, an H-tiling is simply a generalisation of the notion of a matching (which corresponds to the case when H is an edge). An H-tiling in G is perfect if every vertex of G is covered by the H-tiling. Over the last 60 years there have been numerous results on perfect H-tilings. In this talk we give a high-level overview of some of the key ideas that permeate the topic. In particular, we will discuss some typical behaviour of extremal examples, and also some complexity questions.

## Fall 2021

### December 2, 4pm CT

245 Altgeld Hall

**Speaker:** Chandra Checkuri (UIUC Computer Science Dept.)

**Title:** On Submodular k-Partitioning and Hypergraph k-Cut

**Abstract:** Submodular $k$-Partition is the following problem: given a submodular set function $f:2^V \rightarrow \mathbb{R}$ and an integer $k$, find a partition of $V$ into $k$ non-empty parts $V_1,V_2,\ldots,V_k$ to minimize $\sum_{i=1}^k f(V_i)$. Several interesting problems such as Graph $k$-Cut, Hypergraph $k$-Cut and Hypergraph $k$-Partition are special cases. Submodular $k$-Partition admits a polynomial-time algorithm for $k=2,3$ and when $f$ is symmetric also for $k=4$. The complexity is open for $k=4$ and when $f$ is symmetric for $k=5$.

In recent work, motivated by this problem, we examined the complexity of Hypergraph $k$-Cut which only recently admitted a randomized polynomial-time algorithm. We obtained a deterministic polynomial-time algorithm for Hypergraph $k$-Cut as well as new insights in to Graph $k$-Cut. The ideas also led to a polynomial-time algorithm for Min-Max Symmetric Submodular $k$-Partition for any fixed $k$.

The talk will discuss these results with the goal of highlighting the open problem of resolving the complexity of Submodular $k$-Partition.

Based on joint work with Karthik Chandrasekharan

### November 4, 2pm CT

443 Altgeld Hall

**Speaker: **Zoltan Furedi, (Renyi Institute, Hungary and UIUC)

**Title:** Algebraic constructions in combinatorics

**Abstract:** There are two main sources to produce non trivial combinatorial structures. One can use probability theory for typical cases and a bit of algebra for symmetric structures. Here we briefly review classical and new developments and also give examples on how to combine these two powerful methods.

The talk is targeting general mathematicians, with little combinatorics backgrounds.

### October 7, 2021, 4 pm CT

245 Altgeld Hall

**Speaker: **John Shareshian, Washington University of St. Louis

**Title:** A problem on divisors of binomial coefficients, and a theorem on noncontractibility of coset posets

**Abstract:** Fix an integer n>1. It follows directly from a theorem of Kummer that the greatest common divisor of the members of the set BC(n) nontrivial binomial coefficients nC1,nC2,…nC(n-1) is one unless n is a prime power. With this in mind, we define b(n) to be the smallest size of a set P of primes such that every member of BC(n) is divisible by at least one member of P. In joint work with Russ Woodroofe, we ask whether b(n) is at most two for every n. The question remains open.

I will discuss what we know about this question, and how we discovered it during our investigation of a problem raised by Ken Brown about certain topological spaces: Given a finite group G, let C(G) the set of all cosets of all proper subgroups of a finite group, partially ordered by containment. The order complex of C(G) is the simplicial complex whose k-dimensional faces are chains of size k+1 from C(G). We show that this order complex has nontrivial reduced homology in characteristic two, and is therefore not contractible.

If time permits, I will discuss also related work on invariable generation of simple groups, joint with Bob Guralnick and Russ Woodroofe.

### September 30, 2021, 4 pm CT

245 Altgeld Hall

**Speaker: **Bernard Lidicky (Iowa State University)

**Title: ** Flag algebras and its applications

**Abstract: **Flag algebras is a method, developed by Razborov, to attack problems in extremal combinatorics. Razborov formulated the method in a very general way which made it applicable to various settings. The method was introduced in 2007 and since then its applications have led to solutions or significant improvements of best bounds on many long-standing open problems, including problems of Erd\H{o}s. The main contribution of the method was transferring problems from finite settings to limits settings. This allows for clean calculations ignoring lower order terms. The method can utilize semidefinite programming and computers to produce asymptotic results. This is often followed by stability type arguments with the goal of obtaining exact results.

In this talk we will give a brief introduction of the basic notions used in flag algebras and demonstrate how the method works. Then we will discuss applications of the flag algebras in different settings.

**Lunch with the speaker: **Thursday, September 30, 11:45 a.m., Spoon House Korean Kitchen on Green Street; meet at the restaurant.

**Dinner with the speaker:** Thursday, September 30, 6:30 p.m., location to be discussed.

**Additional event:** Friday, October 1, 1:00-1:50 p.m. AH 447; the speakers talk with the students about their favorite problems.

### September 7, 2021, 11-11:50 am CT

347 Altgeld Hall

**Speaker: **Tao Jiang (Miami University of Ohio)

**Title:** Degenerate Turan problems for graphs

**Abstract: **In Turan type extremal problems, we want to determine how dense a graph or hypergraph is without containing a particular subgraph or family of subgraphs. Such problems are central to extremal graph

theory, because solving them requires one to thoroughly investigate the interaction of global graph parameters with local structures. Efforts in solving these problems have spurred the developments of some powerful tools in extremal graph theory, such as the regularity method, probabilistic and algebraic methods.

While Turan problems have satisfactory solutions for non-bipartite graphs, the problem is still generally wide-open for bipartite graphs with many intriguing conjectures and results. In this talk, we will discuss some conjectures on Turan problems for bipartite graphs and some recent progress on them. Time permitting, we will also discuss a colored variant of the Turan problem.

**Additional event: **September 9, 2021, 11-11:50 am CT, 141 Altgeld Hall, Tao Jiang’s favorite problems.