On November 28, Hyunwoo Lee (이현우) from KAIST and IBS Extremal Combinatorics and Probability Group gave a talk at the Discrete Math Seminar on the existence of a Steiner triple system in a 3-uniform hypergraph with large minimum codegree. The title of his talk was “Towards a high-dimensional Dirac’s theorem“.

## Hyunwoo Lee (이현우), Towards a high-dimensional Dirac’s theorem

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.

## Hyunwoo Lee (이현우) gave a talk on partitioning a graph into induced subgraphs each having a subdivision of a fixed graph as a subgraph at the Discrete Math Seminar

On April 25, 2023, Hyunwoo Lee (이현우) from KAIST and IBS Extremal Combinatorics and Probability Group gave a talk at the Discrete Math Seminar about the minimum degree condition to have a partition of a graph into induced subgraphs each having a subdivision of a fixed graph as a subgraph. The title of his talk was “On perfect subdivision tilings.”

## Hyunwoo Lee (이현우), On perfect subdivision tilings

For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $\delta_{sub}(n, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of $\delta_{sub}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is a constant $1 < \xi^*(H)\leq 2$ such that $\delta_{sub}(n, H) = \left(1 - \frac{1}{\xi^*(H)} + o(1) \right)n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise, the threshold depends on the parity of $n$.