Minimalist designs

Abstract

The iterative absorption method has recently led to major progress in the area of (hyper-)graph decompositions. Amongst other results, a new proof of the Existence conjecture for combinatorial designs, and some generalizations, was obtained. Here, we illustrate the method by investigating triangle decompositions: we give a simple proof that a triangle-divisible graph of large minimum degree has a triangle decomposition and prove a similar result for quasirandom host graphs.

Large deviations in random Latin squares

“…It has the same proof as [40,Lemma 2.10]. Let G (3) (n, p) be the random 3-partite 3-graph on the vertex set R ∪ C ∪ S obtained by including all possible edges with probability p independently. Lemma 5.2.…”

“…Let T be a property of unordered partial Latin squares that is monotone increasing in the sense that P ∈ T and P ⊇ P implies P ∈ T . Fix α ∈ (0, 1), let P ∼ L(n, αn 2 ), let G ∼ G (3) (n, α/n) and let G * be the partial Latin square obtained from G by deleting every edge which intersects another edge in more than one vertex (all at once). Then…”

“…In particular, it seems that Theorem 1.2(c) is improvable, but the difficulty lies in finding a general way to complete partial Latin squares to Latin squares without introducing too many intercalates. It seems that Keevash's machinery may not be suitable for this, but the more recent approach of "iterative absorption" due to Glock, Kühn, Lo and Osthus [24] (see also [3]) may be helpful here. • It would be nice to obtain a more accurate understanding of the expected value EN, and to say more about the distribution of N (in particular, it is not even obvious how to estimate the variance of N).…”

Cycle decompositions in $3$-uniform hypergraphs

“…Very recently, Joos and Kühn [15] proved that δ C tends to 1{2 whenever goes to infinity. Together with the best known lower bounds [3,2], we now know that for all odd ě 3,…”

“…Our proof of Theorem 1.1 follows the strategy of iterative absorption introduced by Barber, Kühn, Lo, and Osthus [3] and further developed by Glock, Kühn, Lo, Montgomery, and Osthus [10] to study decomposition thresholds in graphs. We base our outline in the exposition of Barber, Glock, Kühn, Lo, Montgomery, and Osthus [2].…”