기초과학연구원 이산수학그룹
On September 30, 2025, Marcelo Sales from the University of California, Irvine, gave a talk at the Discrete Math Seminar about proving lower bounds on the Ramsey numbers of some hypergraphs. His talk was titled “On the Ramsey number of Daisies and other hypergraphs.”
Given a $k$-uniform hypergraph $H$, the Ramsey number $R(H;q)$ is the smallest integer $N$ such that any $q$-coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$.
When $H$ is a complete hypergraph, a classical argument of Erdős, Hajnal, and Rado reduces the general problem to the case of uniformity $k = 3$. In this talk, we will survey constructions that lift Ramsey numbers to higher uniformities and discuss recent progress on quantitative bounds for $R(H;q)$ for certain families of hypergraphs.
This is joint work with Ayush Basu, Dániel Dobák, Pavel Pudlák, and Vojtěch Rödl.
A subset $A\subseteq \mathbb Z$ of integers is free if for every two distinct subsets $B, B’\subseteq A$ we have \[ \sum_{b\in B}b\neq \sum_{b’\in B’} b’.\]Pisier asked if for every subset $A\subseteq \mathbb Z$ of integers the following two statement are equivalent:
(i) $A$ is a union of finitely many free sets.
(ii) There exists $\epsilon>0$ such that every finite subset $B\subseteq A$ contains a free subset $C\subseteq B$ with $|C|\geq \epsilon |B|$.
In a more general framework, the Pisier question can be seen as the problem of determining if statements (i) and (ii) are equivalent for subsets of a given structure with prescribed property. We study the problem for several structures including $B_h$-sets, arithmetic progressions, independent sets in hypergraphs and configurations in the euclidean space. This is joint work with Jaroslav Nešetřil and Vojtech Rödl.