A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}3\] for all sufficiently large $t$.  Our proof involves a combination of results on the chromatic number of triangle-sparse graphs. Joint work with Jacob Fox and Jonathan Tidor.

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.

For $d\ge 2$ and an odd prime power $q$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb{F}_q$. The distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined to be $\sum_{i=1}^d (x_i-y_i)^2$. An influential result of Iosevich and Rudnev is: if $E \subset \mathbb{F}_q^d$ is sufficiently large and $t \in \mathbb{F}_q$, then there are a pair of points $x,y \in E$ such that the distance between $x$ and $y$ is $t$. Subsequent works considered embedding graphs of distances, rather than a single distance. I will discuss joint work with Debsoumya Chakraborti, in which we show that every sufficiently large subset of $\mathbb{F}_q^d$ contains every nearly spanning tree of distances with bounded degree in each distance. The main novelty in this result is that the distance graphs we find are nearly as large as the set $S$ itself, but even for smaller distance trees our work leads to quantitative improvements to previously known bounds. A key ingredient in our proof is a new colorful generalization of a classical result of Haxell on finding nearly spanning bounded-degree trees in expander graphs. This is joint work with Debsoumya Chakraborti.

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. We consider another natural generalization of the perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6},$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879... + o(1))n$ contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac's theorem.

Given a hypergraph $H=(V,E)$, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:V\to [m]$ such that every hyperedge of $H$ is not monochromatic. The (weak) chromatic number of $H$, denoted by $\chi(H)$, is the smallest $m$ such that $H$ is $m$-colorable. A vertex subset $T \subseteq V$ is called a transversal of $H$ if for every hyperedge $e$ of $H$ we have $T\cap e \ne \emptyset$. The transversal number of $H$, denoted by $\tau(H)$, is the smallest size of a transversal in $H$. The transversal ratio of $H$ is the quantity $\tau(H)/|V|$ which is between 0 and 1. Since a lower bound on the transversal ratio of $H$ gives a lower bound on $\chi(H)$, these two quantities are closely related to each other. Upon my previous presentation, which is based on the joint work with Joseph Briggs and Michael Gene Dobbins (https://www.youtube.com/watch?v=WLY-8smtlGQ), we update what is discovered in the meantime about transversals and colororings of geometric hypergraphs. In particular, we focus on chromatic numbers of $k$-uniform hypergraphs which are embeddable in $\mathbb{R}^d$ by varying $k$, $d$, and the notion of embeddability and present lower bound constructions. This result can also be regarded as an improvement upon the research program initiated by Heise, Panagiotou, Pikhurko, and Taraz, and the program by Lutz and Möller. We also present how this result is related to the previous results and open problems regarding transversal ratios. This presentation is based on the joint work with Eran Nevo.

Determining the density required to ensure that a host graph G contains some target graph as a subgraph or minor is a natural and well-studied question in extremal combinatorics. The celebrated 50-year-old Erdős-Sós conjecture states that for every k, if G has average degree exceeding k-2 then it contains every tree T with k vertices as a subgraph. This is tight as the clique with k-1 vertices contains no tree with k vertices as a subgraph. We present some variants of this conjecture. We first consider replacing bounds on the average degree by bounds on the minimum and maximum degrees. We then consider replacing subgraph by minor in the statement.