## Archive for the ‘2018’ Category

### Jineon Baek, On the off-diagonal Erdős-Szekeres convex polygon problem

Wednesday, August 29th, 2018
On the off-diagonal Erdős-Szekeres convex polygon problem
2018/09/10 Mon 5PM-6PM (Room 1401 of Bldg. E6-1)
The infamous Erdős-Szekeres conjecture, posed in 1935, states that the minimum number ES(n) of points on a plane in general position (that is, no three colinear points) that guarantees a subset of n points in convex position is equal to 2(n-2) + 1. Despite many years of effort, the upper bound of ES(n) had not been better than O(4n – o(n)) until Suk proved the groundbreaking result ES(n)≤2n+o(n) in 2016.
In this talk, we focus on a variant of this problem by Erdős, Tuza and Valtr regarding the number ETV(a, b, n) of points needed to force either an a-cup, b-cap or a convex n-gon for varying a, b and n. They showed ETV(a, b, n) > \sum_{i=n-b}^{a-2} \binom{n}{i-2} by supplying a set of points with no a-cup, b-cap nor a n-gon of that many number, and conjectured that the inequality cannot be improved. Due to their construction, the conjecture is in fact equivalent to the Erdős-Szekeres conjecture. However, even the simplest equality ETV(4, n, n) = \binom{n+1}{2} + 1, which must be true if the Erdős-Szekeres conjecture is, has not been verified yet. To the best of our knowledge, the bound ETV(4, n, n) ≤ \binom{n + 2}{2} – 1, mentioned by Balko and Valtr in 2015, is currently the best bound known in literature.
The talk is divided into two parts. First, we introduce the mentioned works on the Erdős-Szekeres conjecture and observe that the argument of Suk can be directly adapted to prove an improved bound of ETV(a, n, n). Then we show the bound ETV(4, n, n) ≤ \binom{n+2}{2} – C \sqrt{n} for a fixed constant C>0, improving the previously known best bound of Balko and Valtr.

### Hong Liu, Enumerating sets of integers with multiplicative constraints

Friday, August 24th, 2018
Enumerating sets of integers with multiplicative constraints
Hong Liu
Mathematics Institute, University of Warwick, Warwick, UK
2018/9/3 Mon 5PM
Counting problems on sets of integers with additive constraints have been extensively studied. In contrast, the counting problems for sets with multiplicative constraints remain largely unexplored. In this talk, we will discuss two such recent results, one on primitive sets and the other on multiplicative Sidon sets. Based on joint work with Peter Pach, and with Peter Pach and Richard Palincza.

### Tillmann Miltzow, The Art Gallery Problem is ∃R-complete

Tuesday, June 26th, 2018
The Art Gallery Problem is ∃R-complete
Tillmann Miltzow
Computer Science Department, Université Libre de Bruxelles, Brussels
2018/7/24 Tue 5PM-6PM (Room 3434, Bldg. E6-1)
We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p∈P is seen by at least one guard g∈G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃R. The class ∃R consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP⊆∃R. We prove that the art gallery problem is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃R. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many geometric approaches to the problem. This is joint work with Mikkel Abrahamsen and Anna Adamaszek.

### (Intensive Lecture Series) Ron Aharoni, Lectures on topological methods in combinatorics

Wednesday, June 20th, 2018

Intensive Lecture Series

Lectures on topological methods in combinatorics
Ron Aharoni
Department of Mathematics, Technion, Israel
2018/7/17-19 2PM-5PM (Room 3434, Bldg. E6-1)
The lectures will give an introduction to the application of topological methods in matching theory, graph theory, and combinatorics.
Topics that will be covered:

– A topological extension of Hall’s theorem
– combinatorial applications of the nerve theorem
– d-Leray complexes and rainbow matchings
– Matroid complexes and applications
– Open problems

### (Intensive Lecture Series) Jaehoon Kim, A course in graph embedding

Saturday, May 19th, 2018

Intensive Lecture Series

A course in graph embedding
Jaehoon Kim (김재훈)
School of Mathematics, Birmingham University, Birmingham, UK
2018/6/25-29 10:30AM-12PM, 2:30PM-4PM (Room 3434, Bldg. E6-1)

In this lecture, we aim to learn several techniques to find sufficient conditions on a dense graph G to contain a sparse graph H as a subgraph.

Lecture note (PDF file)

Tentative plan for the course (June 25, 26, 27, 28, 29)
Lecture 1 : Basic probabilistic methods
Lecture 2 : Extremal number of graphs
Lecture 3 : Extremal number of even cycles
Lecture 4 : Dependent random choice
Lecture 5 : The regularity lemma and its applications
Lecture 6 : Embedding large graphs
Lecture 7 : The blow-up lemma and its applications I
Lecture 8 : The blow-up lemma and its applications II
Lecture 9 : The absorbing method I
Lecture 10 : The absorbing method II