Date/Time : 08 Apr 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Rao Rui, National University of Singapore

Consider the family of all k-element subsets of {1,…,n}. A subfamily F is called intersecting if any two of its sets have non empty intersection. Huang and Zhao proved that when n>2k, every intersecting family F contains an element i in [n] that belongs to at most {n-2 \choose k-2} members of F. This bound is tight, being attained by 1-star: all k-sets containing a fixed element.

We show that, for n sufficiently large (depending on k), all but at most k+1 elements of [n] have the same property: each of those elements appears in at most {n-2 \choose k-2} sets of F. The number k+1 is tight because of the Hilton-Milner construction of non trivial intersecting family.

This is joint work with Hao Huang.

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Date/Time : 08 Apr 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Ho Boon Suan, National University of Singapore

Given a bipartite graph G with parts of size m and n, Ehrenborg conjectured in 2006 that the number of spanning trees in G is at most the product of all vertex degrees, divided by mn. We will discuss a linear-algebraic proof of this bound, including a characterization of when equality holds.

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Date/Time : 18 Mar 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Fan Chang, Nankai University

In this talk, I will present a short and elementary proof of the Poincaré inequality for increasing subsets of the hypercube, recently proved by Fei and Ferreira Pinto Jr. This result implies an $O(n^2)$ upper bound on the mixing time of censored random walks, improving on the earlier $O(n^3)$ bound of Ding and Mossel. This is joint work with Guowei Sun and Lei Yu.

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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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