Maria-Romina Ivan, University of Cambridge

Abstract

A poset is short for a partially ordered set. The most common example of a poset is the power set of $[n]$ with the partial relation given by inclusion. Given a fixed poset $mathcal P$, we say that a family $mathcal F$ of subsets of $ [n]$ is $mathcal P$-free if there is no induced copy of $mahcal P$ formed by elements of $mathcal F$. We further say that $mathcal F$ is $mathcal P$-saturated if it is $mathcal P$-free and, for any other set $X$ not in $mathcal F$, the family formed by adding $X$ to $mathcal F$ contains an induced copy of $mathcal P$. The size of the smallest $mathcal P$-saturated family is called the induced saturation number of $mathcal P$.

The natural question is: what can we say about the saturation number? Even for simple posets such as the the antichain and the butterfly, the question has proved difficult – for the diamond poset the question is surprisingly wide open.

How about the saturation number for an arbitrary poset $mathcal P$? Freschi, Piga, Sharifzadeh and Treglown proved that the saturation number for any poset is either bounded, or at least $sqrt n$. What about the upper bound? Or, can we characterise the posets that have unbounded saturation number?

In this talk we will discuss recent developments, most notably a wide class of posets that have unbounded saturation number, as well as a general polynomial upper bound.

See: https://sites.google.com/view/rutgersdmseminar