The ACO Seminar (2025–2026)

In 1847, Kirkman proved that there exists a Steiner triple system on n vertices (equivalently a triangle decomposition of the edges of K_n) whenever n satisfies the necessary divisibility conditions (namely n is congruent to 1 or 3 mod 6). In 1970, Nash-Williams conjectured that every graph G on n vertices with minimum degree at least 3n/4 (for n large enough and G satisfying the necessary divisibility conditions) has a triangle decomposition. Nash-Williams’ Conjecture is a central open question in extremal design theory. Here we discuss recent progress on the conjecture and its generalizations. Joint work with Michelle Delcourt, Cicely Henderson, and Thomas Lesgourgues.