If a partial order on $n$ elements has $m$ incomparable pairs, then its covering graph (aka Hasse diagram aka transitive reduction, the graph of pairs of elements that are comparable but are not the endpoints of a chain of three or more elements) can only have $O(n+m)$ edges, and this is tight — see http://11011110.livejournal.com/233793.html for a proof.

I tried doing some literature searches but the relevant keywords are broad and the topic itself is rather specific, so I didn't find it. (I did find some papers on cover-incomparability graphs, the unions of the covering graph and the incomparability graph.) Is this known, and if so, can someone provide a reference please?