報告人：魯紅亮 教授
報告題目：Towards the Erd\H{o}s matching conjecture for 4-uniform hypergraphs: stability and applications
報告時間：2026年5月20日（周三）下午7：30
報告地點(diǎn)：騰訊會議：907-381-1970
主辦單位：數(shù)學(xué)與統(tǒng)計學(xué)院、數(shù)學(xué)研究院、科學(xué)技術(shù)研究院
報告人簡介：
魯紅亮，西安交通大學(xué)教授，博士生導(dǎo)師，研究方向組合最優(yōu)化和極值圖論，主持多項(xiàng)國家基金項(xiàng)目，發(fā)表論文百余篇。獲邀第十三屆海峽兩岸圖論與組合數(shù)學(xué)大會報告（2025）、國際基礎(chǔ)科學(xué)大會邀請報告(ICBS)（2023）、華人數(shù)學(xué)家大會邀請報告(ICCM)（2022）等國際重要大會報告。榮獲第十三屆陜西青年科技獎（2020），入選國家級青年人才項(xiàng)目---四青人才計劃（2019），中國運(yùn)籌學(xué)會圖論與組合分會青年論文獎一等獎（2017）等。
報告摘要：
A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$.This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author.
In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form. This is joint work with Frankl, Ma, and Wu.