报告题目：Recent Developments in the Brown--Erd{\H{o}}s--S{\'o}s Problem
报告人：黄欣祺博士后
报告时间：2026年9月20日16:00-17:00
报告地点：科技园阳光楼南815
邀请单位：福州大学离散数学研究中心/数学与统计学院
报告内容简介：
Hypergraph Tur\'an problems are central objects in extremal combinatorics. A celebrated example is the Brown--Erd{\H{o}}s--S{\'o}s problem. For integers $r,k\ge 2$ and $r\le s\le rk$, let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. Brown, Erd{\H{o}}s, and S{\'o}s established the general bounds $\Omega\bigl(n^{(rk-s)/(k-1)}\bigr)\le f^{(r)}(n;s,k)\le O\bigl(n^{\lceil(rk-s)/(k-1)\rceil}\bigr)$. One important direction is the classical conjecture that $f^{(3)}(n;k+3,k)=o(n^2)$. When $k=3$, this is the celebrated $(6,3)$-problem, solved by Ruzsa and Szemer\'edi using the triangle removal lemma, whereas already the next case, the $(7,4)$-problem, remains open.
Another fundamental direction in the Brown--Erd{\H{o}}s--S{\'o}s problem concerns the integral-exponent regime $s=(r-t)k+t$, where $2\le t<r$; this will be the main focus of the talk. In this regime, the general bounds give $f^{(r)}\bigl(n;(r-t)k+t,k\bigr)=\Theta(n^t)$, so the Tur\'an exponent is already known. The central question is therefore whether the normalized leading coefficient $\pi(r,t,k):=\lim_{n\to\infty}n^{-t}f^{(r)}\bigl(n;(r-t)k+t,k\bigr)$ exists and, if so, what its value is. In recent work, we settle the existence question throughout this regime and determine the coefficient except when $(r,t)=(3,2)$ and $k\ge 4$ is even. Outside this exceptional family, the answer depends on $k$ only through its parity, while the natural lower bound is not sharp in the exceptional case.
I will survey the historical development and basic methods of the problem, with particular emphasis on the lower-bound constructions and their connections to combinatorial design theory. This is joint work with Ting-Wei Chao and Hong Liu.
报告人简介：
黄欣祺 中国科学技术大学博士生，目前在IBS ECOPRO长期访问。导师: 刘鸿教授，张先得教授，Tuan Tran教授。感兴趣的方向为极值图论、编码理论和加性组合。
