Seminar Details

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.