Date/Time : 29 Mar 2023 16:00
Venue : via Zoom
Speaker : Xizhi Liu, University of Warick

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Meeting ID: 810 4646 2775
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We present a general approach for determining the maximum number of edges in an n-vertex r-uniform hypergraph that does not contain t+1 vertex-disjoint copies of F, when n is large, t = o(n), and F is nondegenerate. This extends old results of Simonovits and Moon on complete graphs.

Based on joint work with Jianfeng Hou, Heng Li, Long-Tu Yuan, and Yixiao Zhang.

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Date/Time : 22 Mar 2023 15:00
Venue : via Zoom
Speaker : Huy Tuan Pham, Stanford University

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Meeting ID: 810 4646 2775
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A graph is Ramsey if its largest clique or independent set is of size logarithmic in the number of vertices. While almost all graphs are Ramsey, there is still no known explicit construction of Ramsey graphs. We discuss recent progress on finding these “dark matter” of graphs. Along the way, we study some fundamental problems in additive combinatorics, and discover that group structure is superfluous for these problems. Joint work with David Conlon, Jacob Fox, and Liana Yepremyan.

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Date/Time : 15 Mar 2023 10:00
Venue : via Zoom
Speaker : Cosmin Pohoata, Institute for Advanced Study

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Meeting ID: 810 4646 2775
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Suppose we are given matchings M_1,….,M_N of size t in some r-uniform hypergraph, and let us think of each matching having a different color. How large does N need to be (in terms of t and r) such that we can always find a rainbow matching of size t? This problem was first introduced by Aharoni and Berger, and has since been studied by several different authors. While interesting for its own sake in the context of finding transversals in Latin Squares (e.g. the Ryser–Brualdi–Stein Conjecture), it is perhaps not too surprising that this question is also connected with different sides of additive combinatorics. In this talk, we will survey some of these connections, discuss some recent results that more or less completely answer the original question when one of the parameters t or r is fixed, and then end with some (fairly arbitrary) speculations/open problems. Joint work with Lisa Sauermann and Dmitrii Zakharov.

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Date/Time : 01 Mar 2023 15:00
Venue : via Zoom
Speaker : Shukun Wu, California Institute of Technology

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Meeting ID: 810 4646 2775
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The Kakeya conjecture suggests a fundamental phenomenon in Euclidean space: (thin) tubes with different directions are essentially disjoint. The problem also arises naturally in other contexts, (e.g. Fourier analysis, geometric measure theory), and it often serves as a main enemy—tool. In this talk, I will briefly survey the conjecture and some techniques people developed to attack it. Along the way, I will also discuss some recent progress and applications.

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Date/Time : 15 Feb 2023 16:00
Venue : via Zoom
Speaker : Akshat Mudgal, University of Oxford

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Let d be a fixed positive integer and let A be a finite set of n elements in R^d. A classical result of Freiman then suggests that if A is not contained in a translate of some hyperplane, then

|A+A| >= (d+1)n – d(d+1)/2,

where A+A = {a + b : a,b in A}. This is known to be sharp by setting A to be a union of arithmetic progressions. Naturally, one may ask whether a corresponding result holds for the difference set A-A = { a – b : a,b in A}. This forms a conjecture of Uhrin from 1980, which was subsequently raised again by Ruzsa in 1994. While the d=1,2 and d=3 cases of this problem were solved by Freiman-Heppes-Uhrin (1989) and Stanchsecu (1998) respectively, the d> 3 case was open till quite recently. In this talk, we will survey some of the recent results in this direction, including our own work which shows that whenever d>3 and A is not lying in some affine hyperplane, we have

|A-A| >= (2d – 2 + 1/(d-1))n – O(n^{1-c}),

for some constant c = c(d) >0. The main term here is sharp, thus resolving the conjecture of Uhrin and Ruzsa up to a polynomially small error term.

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Date/Time : 09 Nov 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Debsoumya Chakraborti, Institute for Basic Science

Abstract:  There have been extensive studies on the following question: given k graphs G_1,… , G_k over a common vertex set of size n, what conditions on G_i ensure a rainbow copy of H, i.e., a copy of H with at most one edge from each G_i? In this talk, we will survey some results on this topic.

Joos and Kim showed that for k=n, the condition that each G_i has minimum degree at least n/2 confirms a rainbow Hamiltonian cycle. This generalizes the classical Dirac’s theorem. In this talk, we will discuss our recent rainbow generalizations of some results on embedding Hamiltonian cycles in tournaments. This part of the talk will be based on a joint work with Jaehoon Kim, Hyunwoo Lee, and Jaehyeon Seo.

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Date/Time : 02 Nov 2022 14:00
Venue : via Zoom
Speaker : Zilin Jiang, Arizona State University

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Meeting ID: 816 3960 4748
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A set of unit vectors in a Euclidean space is called a spherical two-distance set if the pairwise inner products of these vectors assume only two values α and β. It is known that the maximum size of a spherical two-distance grows quadratically as the dimension of the Euclidean space grows. However when the values α and β are held fixed, a very intricate behavior of the maximum size emerges. Building on our recent resolution in the equiangular case, that is β = -α, we make a plausible conjecture which connects this behavior with spectral theory of signed graphs in the regime β < 0 ≤ α, and we confirm this conjecture when α + 2β < 0 or (1 – α)/(β-α) < 2.0198. Joint work with Alexandr Polyanskii, Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao.

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Date/Time : 26 Oct 2022 14:00
Venue : via Zoom
Speaker : Yuval Wigderson, Tel Aviv University

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Meeting ID: 896 1391 3454
Passcode: 23571113

Abstract: The Ramsey number r(G) of a graph G is the minimum N so that every red/blue edge-coloring of K_N contains a monochromatic copy of G. Conlon, Fox, and Sudakov conjectured that if we delete a single vertex from G, then the Ramsey number can decrease by at most a constant factor. Though very natural (and true in a variety of special cases), this conjecture turns out to be false in general. In this talk, I’ll explain how one disproves this conjecture, as well as the connections this problem has to a number of other questions in Ramsey theory.

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Date/Time : 19 Oct 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Yang Tianchi, National University of Singapore

Abstract: The study of the 4-cycle has an important enlightening effect on the development of Turan type problems, especially the degenerate cases. In this talk, we focus on two conjectures about 4-cycles: a conjecture of Erdos on the upper bound of the Turan number ex(n,C_4), and a conjecture of Erdos-Simonovits on the supersaturation problem of 4-cycles.

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