My previous post was about an asymptotic solution of Rota’s basis conjecture and the next few posts will also be devoted to some mathematical news. Before moving to the main featured result let me mention a beautiful blog post by Maria Gillespie about Tatsuyuki Hikita’s startling proof of the Stanley-Stembridge conjecture. Hikita’s proof was also recently featured in the facebook feed of Raul Basu (that contains a lot of very interesting news about mathematics and physics.)
Poset saturation
The study of post saturation was initiated in a 2013 paper by Dániel Gerbner, Balázs Keszegh, Nathan Lemons, Cory Palmer, Dömötör Pálvölgyi, and Balázs Patkós. The new result has been the big open problem in poset saturation.
Maria-Romina Ivan and Sean Jaffe: The saturation number for the diamond is linear
Abstract: For a fixed poset
we say that a family
is
-saturated if it does not contain an induced copy of
, but whenever we add a new set to
, we form an induced copy of
. The size of the smallest such family is denoted by
. For the diamond poset
(the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most
, the best known lower bound has stayed at
since the introduction of the area of poset saturation. In this paper we prove that
, establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems which may be of independent interest.
we say that a family ![\mathcal F\subseteq\mathcal P ([n])](http://s0.wp.com/latex.php?latex=%5Cmathcal+F%5Csubseteq%5Cmathcal+P+%28%5Bn%5D%29&bg=ffffff&fg=333333&s=0&c=20201002)
is 
, we form an induced copy of 
. For the diamond poset 
(the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most 
, the best known lower bound has stayed at 
since the introduction of the area of poset saturation. In this paper we prove that 
, establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems which may be of independent interest.
Short commentary:
Even for very small posets (like the diamond, alias the two-dimensional Boolean lattice), finding the growth of the saturation number is very difficult. A maximal chain shows that the saturation number for the diamond is at most n+1. Lots of people have worked on showing that the saturation number is linear —indeed, this was one of the main focus-problems for a whole workshop in Budapest, namely the workshop on “The Forbidden Poset Problem”. There had been lots of small bits of progress, culminating in a lower bound of 
Imre Leader wrote to me about the new result:
“This was kind of the `holy grail’ for poset saturation. Well, it was the holy grail for specific posets. There are still the wonderful general conjectures that nobody can solve, namely
- Is the saturation number for every poset either constant or at least linear?
- Is the saturation number for every poset at most linear?
For a summary of the state of the art on those two conjectures see Gluing Posets and the Dichotomy of Poset Saturation Numbers, by Maria-Romina Ivan and Sean Jaffe.”
Let me add that the notions of saturation and weak saturation of graphs and hypergraphs have led to beautiful research in extremal and probabilistic combinatorics. There are many interesting open problems and exciting recent results. Bollobas’s Two Families theorem (discussed in this post) gives the saturation number of complete 
h/t Imre Leader