Two related papers will be discussed: 1. In 1966, Erdős, Goodman, and Pósa showed that if $G$ is an $n$-vertex graph, then at most $\lfloor n^2/4 \rfloor$ cliques of $G$ are needed to cover the edges of $G$, and the bound is best possible as witnessed by the balanced complete bipartite graph. This was generalized independently by Győri–Kostochka, Kahn, and Chung, who showed that every $n$-vertex graph admits an edge-decomposition into cliques of total `cost’ at most $2 \lfloor n^2/4 \rfloor$, where an $i$-vertex clique has cost $i$. Erdős suggested the following strengthening: every $n$-vertex graph admits an edge-decomposition into cliques of total cost at most $\lfloor n^2/4 \rfloor$, where now an $i$-vertex clique has cost $i-1$. We prove fractional relaxations and asymptotically optimal versions of both this conjecture and a conjecture of Dau, Milenkovic, and Puleo on covering the $t$-vertex cliques of a graph instead of the edges. Our proofs introduce a general framework for these problems using Zykov symmetrization, the Frankl–Rödl nibble method, and the Szemerédi Regularity Lemma. It is joint work with Jialin He, Robert Krueger, The Nguyen, and Michael Wigal. 2. Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Győri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Turán graph on $n$ vertices and $r – 1$ parts, then $G$ has at least $(2 – o(1))k/r$ edge-disjoint $r$-cliques. We prove this conjecture. It is joint work with Michael Wigal.

A backedge graph of a tournament $T$ with respect to a total ordering $\prec$ of the vertices of $T$ is a graph on $V(T)$ where $uv$ is an edge if and only if $uv \in A(T)$ and $v \prec u$. In 2023, Aboulker, Aubian, Charbit and Lopes introduced the clique number of tournaments based on backedge graphs as a natural counterpart to the dichromatic number of tournaments. Specifically, the clique number of a tournament is the minimum clique number of a backedge graph when considering all possible orderings. Given this definition, it is not immediately clear what the canonical clique object should be. In this talk, we provide an answer to this question. We show that if a tournament has large clique number, it contains a reasonably large subtournament from one of two simple and previously studied families of tournaments of unbounded clique number. This talk is based on joint work with Logan Crew, Xinyue Fan, Benjamin Moore, and Sophie Spirkl.

We overview the recent resolution of a 1985 open problem of Gyárfás, that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path. The proof introduces a chromatic density framework involving chromatic quasirandomness and chromatic density increment, which allows us to deduce the desired statement from the Erdős–Hajnal result for the five-vertex path.

Separating hash families are useful combinatorial structures that generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of the universe for a $t$-perfect hash family of length $N$ over an alphabet of size ( q ). We show that $q^{2 – o(1)} < p_t(t, q) = o(q^2)$ for all  $t \ge 3$, thereby resolving an open problem raised by Blackburn et al. (2008) for certain parameter ranges. Previously, this result was known only for $t = 3$ and $t = 4$. Our approach establishes the existence of a large set of integers that avoids nontrivial solutions to a system of correlated linear equations. This is joint work with Xiande Zhang and Gennian Ge.

Directed graphs prove to be very hard to tame in contrast to undirected graphs. In particular, they are not well-quasi-ordered by any known relevant inclusion relation, and are lacking fruitful structure theorems. This motivates the search for structurally rich subclasses of directed graphs that are well behaved. Eulerian directed graphs are a particularly prominent example, sharing many similarities with undirected graphs. In fact, it is conjectured that Eulerian directed graphs are well-quasi-ordered by weak immersion, and even well-quasi-ordered by strong immersion when restricting to classes of bounded degree. We believe that we have a proof of both conjectures, and I will report on the current status, progress, and steps towards said proof and its implications. This is joint work with Ken-ichi Kawarabayashi and Stephan Kreutzer.

A k-coloring of a graph is an assignment of k colors to its vertices such that no two adjacent adjacent vertices receive the same color. The Coloring Problem is the problem of determining the smallest k such that the graph admits a k-coloring. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. The complexity of the Coloring Problem for L-free graphs is known (NP-complete or polynomial-time solvable) whenever L contains a single graph. There has been keen interest in coloring graphs whose forbidden list L contains basic graphs such as induced paths, induced cycles and their complements. In this talk, I will provide a survey of recent progress on this topic.