Consider a general Turan-type problem on hypergraphs. Let $\mathcal{F}$ be a family of $k$-subsets of $[n]$ that does not contain sets $F_1, \ldots, F_s$ satisfying some property $P$. We show that if $P$ is low-dimensional in some sense (e.g., is defined by intersections of bounded size) then, under polynomial dependencies between $n, k$ and the parameters of $P$, one can reduce the problem of maximizing the size of the family $|\mathcal{F}|$ to a finite extremal set theory problem independent of $n$ and $k$. We show that our technique implies new bounds in a number of Turan-type problems including the Erdős-Sós forbidden intersection problem, the Duke-Erdős forbidden sunflower problem, forbidden $(t, d)$-simplex problem and the forbidden hypergraph problem. Furthermore, we also briefly discuss the connection between the aforementioned reduction and the measure boosting argument based on the action of a certain semigroup on the Boolean cube.  This connection turns out to be fruitful when extending extremal set theory problems to domains different from $\binom{[n]}{k}$. Joint work with Liza Iarovikova, Andrey Kupavskii, Georgy Sokolov and Nikolai Terekhov

Given a $k$-uniform hypergraph $F$, its Turán density $\pi(F)$ is the infimum over all $d\in [0,1]$ such that any $n$-vertex $k$-uniform hypergraph $H$ with at least $d\binom{n}{k}+o(n^k)$ edges contains a copy of $F$. While Turán densities are generally well understood for graphs ($k=2$), the problem becomes notoriously difficult for $k\geq 3$, even for small hypergraphs. We study two well-known variants of this Turán problem for hypergraphs: first, under minimum codegree conditions and, second, with a quasirandom edge distribution. Each variant defines a distinct extremal parameter, generalising the classical Turán density. Here we present recent results in both settings, with a particular emphasis on the case of hypergraphs where every link is itself quasirandom. Our results include exact solutions for key hypergraphs and general results about the behaviour of the Turán density functions.

We consider a continuous model of graphs, introduced by Dearing and Francis in 1974, where each edge of G to be a unit interval, giving rise to an infinite metric space that contains not only the vertices of G but all points on all edges of G. Several standard graph problems can be defined and studied on continuous graphs, yielding many surprising algorithmic results and combinatorial connections. The motivation can be exemplified by the well-known Independent Set problem on graphs. Given a graph G, we want to place k facilities that are pairwise at least a distance 2 edge lengths apart. In some applications, such as when the underlying graph represents a street network, it is reasonable to allow placing a facility not only at a crossroad but also somewhere within a street, that is, not only at a vertex but also at any point on an edge between two vertices. This motivates the study of the Independent Set problem on continuous graphs. In such a setting, for example, the problem corresponds to requiring pairwise distance r=2 for the placed facilities. However, we may also study the problem where we fix r to any positive integer, rational, or even irrational number. Other problems studied in the continuous model include Dominating Set, TSP, and Coloring. In the first part of this talk, we will give a general overview of research on continuous graphs and computational problems in this setting. In the second part, we explore, as part of recent work, a coloring problem on continuous graphs akin to the well-known Hadwiger-Nelson Problem. Based on joint work with Fabian Frei, Florian Hörsch, Tom Janßen, Stefan Lendl, Dániel Marx, Prahlad Narasimhan, and Gerhard Woeginger.