I'm trying to characterise hypergraphs $H=(V,E)$ such that there exists a linear order $<$ of $V$ which satisfies $\small\forall X,Y\in E[X\subset Y\implies \max(X)<\max(Y)]$?. Are there a few 'elementary' criteria on $H$ that I can use to determine when such an order exists?
I'm thinking maybe I could transform the problem into determining when a certain poset is ranked.
Assuming that $E$ is a family of nonempty finite subsets of $V$, a necessary and sufficient condition for the existence of such a linear order is that there is no nonempty finite set $W\subseteq V$ such that, for each vertex $w\in W$, there are hyperedges $X,Y\in E$ such that $w\in X\subsetneqq Y\subseteq W$.
The necessity is obvious; if $W$ is such a set, then for any linear order there are hyperedges $X,Y\in E$ with $X\subsetneqq Y$ and $\max(X)=\max(Y)=\max(W)$.
In proving the sufficiency, we may assume that $V$ is finite; the general case will then follow by the usual sort of compactness argument.
Suppose that $V$ is an $n$-element set, and that every nonempty subset $W\subseteq V$ contains an element $w$ such that $\{X\in E:w\in X\subseteq W\}$ is an antichain. If $i\in\{1,\dots,n\}$ and if $w_1,\dots,w_{i-1}$ have already been chosen, let $W_i=V\setminus\{w_1,\dots,w_{i-1}\}$, and choose $w_i\in W_i$ so that $\{X\in E:w\in X\subseteq W_i\}$ is an antichain. Finally, define the linear order $\lt$ so that $w_1\gt\cdots\gt w_n$, i.e., $w_i=\max(W_i)$.