# Uniform three-class regular partial Steiner triple systems with uniform degrees

A Partial Steiner Triple system $(\mathcal{X}, \mathcal{T})$ is a finite set of points $\mathcal{X}$ and a collection $\mathcal{T}$ of $3$-element subsets of $\mathcal{X}$ that every pair of points intersect in at most $1$ triple. A $3$-class regular PSTS (written as $3$-PSTS([$m \cdot \alpha, n\cdot \beta, p \cdot \gamma$])) is a PSTS where the points can be partitioned into $3$ classes (each class having size $m$, $n$ and $p$ respectively) such that no triple belongs to any class and any two points from the same class occur in the same number of triples ($\alpha$, $\beta$ and $\gamma$ respectively). The $3$-PSTS is said to be uniform if $m = n = p$. In this presentation, I will talk about the existence of uniform $3$-PSTS with uniform degrees ($\alpha = \beta = \gamma$).