A family of sets in $[n]$ is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. The problem was completely solved when $\ell$ is a prime via an elegant linear algebraic method, showing that the family has size at most $n$. However, not much was known for composite numbers. By splitting the family into families correspond to each prime factor of $\ell$, one can show that the number is at most $\omega n$, where $omega$ is the number of prime factors of $\ell$. We used both combinatorial and Fourier analytic arguments to prove that the number of sets in any $\ell$-Oddtown is at most $\omega n-(2\omega+\varepsilon)\log_2 n$ for most $n,\ell$.
Ting-Wei Chao (趙庭偉) gave a talk on the number of points that are intersections of d linearly independent lines among given n lines in the d-dimensional space at the Discrete Math Seminar
On December 12, 2023, Ting-Wei Chao (趙庭偉) from Carnegie Mellon University gave a talk at the Discrete Math Seminar on the number of points that are intersections of d linearly independent lines among given n lines in the d-dimensional space. The title of his talk was “Tight Bound on Joints Problem and Partial Shadow Problem“.
Ting-Wei Chao (趙庭偉), Tight Bound on Joints Problem and Partial Shadow Problem
Given a set of lines in $\mathbb R^d$, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most $O(N^{3/2})$ joints in $\mathbb R^3$ via the polynomial method.
Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollobás and Eccles on the partial shadow problem. It is surprising to us that the only known proof of this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.