A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular,
uniform hypergraph HH with small maximum codegree has an almost‐perfect matching. We …

We solve the existence problem for $ F $-designs for arbitrary $ r $-uniform hypergraphs $ F
$. This implies that given any $ r $-uniform hypergraph $ F $, the trivially necessary …

We generalise the existence of combinatorial designs to the setting of subset sums in lattices
with coordinates indexed by labelled faces of simplicial complexes. This general framework …

We prove the existence of block designs in complexes and hypergraphs whose clique
distribution satisfies appropriate regularity constraints. As a special case, this gives a new …

A fundamental theorem of Wilson states that, for every graph F, every sufficiently large F-
divisible clique has an F-decomposition. Here a graph G is F-divisible if e (F) divides e (G) …

We prove that for $ n\in\mathbb N $ and an absolute constant $ C $, if $ p\geq C\log^ 2 n/n $
and $ L_ {i, j}\subseteq [n] $ is a random subset of $[n] $ where each $ k\in [n] $ is included …

We solve the existence problem for $ F $-designs for arbitrary $ r $-uniform hypergraphs $ F
$. In particular, this shows that, given any $ r $-uniform hypergraph $ F $, the trivially …

We study the F-decomposition threshold δ F for a given graph F. Here an F-decomposition of
a graph G is a collection of edge-disjoint copies of F in G which together cover every edge of …

Our main result essentially reduces the problem of finding an edge-decomposition of a
balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a …

Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition
of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently …