Chouinard's Conjecture for Graphical t-Designs

Abstract

A $t$-wise balanced design on the edge set of a complete graph is graphical if its block multiset is invariant under the induced action of the symmetric group on the vertices. Chouinard conjectured that, for each fixed index $λ$, there are only finitely many nontrivial simple graphical $t$-wise balanced designs with $t>1$. We prove the conjecture for $t$-designs, which are the $t$-wise balanced designs whose blocks all have the same size. Our theorem does not require simplicity, and we bound all parameters of the designs by explicit polynomials in $λ$.