Date/Time : 11 Feb 2026 16:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Jing Yifan, Ohio States University

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $\Omega(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets.

We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lov\’asz-Sauermann. Joint work with Cosmin Pohoata and Max Wenqiang Xu.

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Date/Time : 06 Feb 2026 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Fang Yijia, National University of Singapore

Euclidean Ramsey theory asks if we can find specific geometric structures in colored space. A refined version of this, known as the canonical Ramsey property, asks: Does every finite coloring of Euclidean space contain a congruent copy of a shape that is either monochromatic or rainbow (all in different colors)?

While this was known for specific shapes, such as acute and right triangles, the question of whether all triangles satisfy this property remained open. In this talk, I will present a solution to this problem posed by Gehér, Sagdeev, and Tóth. We prove that 4-dimensional space is sufficient to guarantee the canonical property for any triangle. The proof uses rotation arguments and a particular configuration that force monochromatic structures.

(Joint work with Gennian Ge, Yang Shu, Qian Xu, Zixiang Xu, and Dilong Yang.)

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Date/Time : 28 Jan 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Swee Hong Chan, Rutgers University

Consider the set of positive integers representing the number of spanning trees in simple graphs with n vertices. How quickly can this set grow as a function of n? In this talk, we discuss a proof of the exponential growth of this set, which resolves an open problem of Sedlacek from 1966. The proof uses a connection with continued fractions and advances towards Zaremba’s conjecture in number theory. This is joint work with Alex Kontorovich and Igor Pak. This talk is intended for general audience.

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Date/Time : 16 Jan 2026 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Seog-Jin Kim, Konkuk University, Korea

The square of a graph G, denoted G^2, has the same vertex set as G and has an edge between two vertices if the distance between them in G is at most 2. Thomassen (2018) showed that \chi(G^2)<=7 if G is a subcubic planar graph. A natural question is whether \chi_l(G^2)<=7 or not if G is a subcubic planar graph. Recently Kim and Lian (2024) showed that \chi_l(G^2)<=7 if G is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim (2025) showed that \chi_l(G^2)<=7 if G is a subcubic planar graph without 4-cycles and 5-cycles. In this talk, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the previous results.

This is joint work with Xiaopan Lian, Atsuhiro Nakamoto, and Kenta Ozeki.

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Date/Time : 05 Nov 2025 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Ander Lamaison, Institute for Basic Science, South Korea

We study Turán problems for hypergraphs with an additional uniformity condition on the edge distribution. This kind of Turán problems was introduced by Erdős and Sós in the 1980s but it took more than 30 years until the first non-trivial exact results were obtained. Central to the study of the uniform Turán density of hypergraphs are palette constructions, which were implicitly introduced by Rödl in the 1980s. We prove that palette constructions always yield tight lower bounds, unconditionally confirming present empirical evidence. This results in new and simpler approaches to determining uniform turán densities, which completely bypass the use of the hypergraph regularity method.

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Date/Time : 17 Oct 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yibo Gao, BICMR, Peking University

A partially ordered set is called Sperner if the size of its largest antichain does not exceed the size of its largest rank. In this talk, we show that the weak Bruhat order of the symmetric group is strongly Sperner via sl2 representations, resolving an old conjecture by Björner. We will also discuss its relations with Schubert polynomials, which are polynomial representations of the Schubert classes in the cohomology ring of the flag variety.

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Date/Time : 03 Oct 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Mihir Mittal, Indian Institute of Technology Kanpur

Zarankiewicz’s problem asks: Given a bipartite graph on n vertices without a K_{k,k} subgraph, how many edges can it have? The classical Kővári–Sós–Turán Theorem gives us the bound O(n^{2-1/k}); this remains the best possible bound for general situations. In geometric settings, however, stronger bounds can be achieved. Recent work of Basit, Chernikov, Starchenko, Tao, and Tran showed that semi-linear incidence graphs in real vector spaces admit almost-linear Zarankiewicz bounds.

In this talk I will describe a counterpart of their result for the additive group of the p-adic numbers and more generally in valued vector spaces. This is obtained through introducing a new purely combinatorial notion of laminarity. As a corollary, we get almost-linear bounds for K_{k,k}-free point–box and point–polytope incidence graphs in these settings.

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Date/Time : 29 Jul 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Vilas Sreenivasan Winstein, University of California, Berkeley

Independent sets have long been studied as a model for gases in statistical mechanics, and Sapozhenko’s famous graph container method was developed to understand their quantity in the hypercube, where they correspond to error-detecting codes. When removing edges at random (i.e. percolating the hypercube), the number of independent sets in the resulting graph becomes the partition function of a disordered system, akin to popular models like spin glasses. I will discuss a new probabilistic framework which establishes sharp distributional characterizations of this partition function and the underlying disordered system, settling several open problems and explaining various phase transitions observed in this model. This is joint work with Mriganka Basu Roy Chowdhury and Shirshendu Ganguly.

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Date/Time : 17 Apr 2025 13:00
Venue : S16 #03-06
Speaker : Abdul Basit, Monash University

The independent set sequence of a graph is the sequence whose k^th term is the number of independent sets of size k in the graph. Alavi, Malde, Schwenk and Erdős studied this sequence in 1987, and showed that it can exhibit any pattern of rises and falls. They raised the question, however, of whether the independent set sequence of a tree, or more generally a forest, is always unimodal — that is, has a single peak. This intriguing question is still very much open. I’ll survey some recent results about this question. (Some of this is joint work with David Galvin and Misha Isarev.)

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