Let $G = (V, E)$ be a graph on $n$ vertices, and let $c : E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. We show that for sufficiently large graphs $G$, the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path. We also show sufficient conditions on $\delta^c(G)$ for the existence of a rainbow cycle of length $2k$ in sufficiently large bipartite graphs $G$, which are tight in many cases. This is joint work with Andrzej Czygrinow.

Flag algebras are a mathematical framework introduced by Alexander Razborov in 2007, which has been used to resolve a wide range of open problems in extremal graph theory in the past twenty years. This framework provides an algebraic setup where one can express relationships between induced subgraph densities symbolically. It also comes with mathematical techniques for systematically deriving such relationships that always hold. Some of these techniques have even been implemented in automatic tools, such as Flagmatic. In this work, we formalise flag algebras in Lean, an interactive theorem prover based on dependent type theory. Lean is computer software that lets us write mathematical definitions and proofs in a computer and check the correctness of written proofs using a computer. By formalizing flag algebras in Lean, we can ensure that the mathematical foundation of flag algebras does not have any mistakes, and provide a mathematical guarantee that theorems proved by flag algebras are indeed correct. In this talk, I will explain flag algebras and our Lean formalization using Mantel’s theorem as a guiding example. In the process, I will describe the challenges and lessons learned during the formalization. This is a joint work with Jihoon Hyun, Gyeongwon Jeong, Sang-il Oum, and Hongseok Yang.