On a Combinatorial Theorem of Curtis Greene

On Greene-Kleitman's theorem for general digraphs

Aharoni, R. and I. Ben-Arroyo Hartman, On Greene-Kleitman’s theorem for general digraphs, Discrete Mathematics 120 (1993) 13-24. Linial conjectured that Greene-Kleitman’s theorem can be extended to general digraphs. We prove a stronger conjecture of Berge for digraphs having k-optimal path partitions consisting of ‘long’ paths. The same method yields known results for acyclic digraphs, and exte...

A Combinatorial Theorem on Systems of Sets

It is quite easy to see that (2) and (3) give best possible results. Upper estimates for n were also given in [2] if either (Alt..., An)eS(k, </, m) and /<^(m+l) or if (Alt..., An)eS(k, I, m) and l^^(m+k), but the results in these cases are not best possible. Other results of this kind have been established in [3] and [4] which settle certain conjectures made in [2]. In this note we establish t...

A Quasi Curtis-Tits-Phan theorem for the symplectic group

We obtain the symplectic group Sp(V ) as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let Sp(V ) act flag-transitively on the geometry of maximal rank subspaces of V . We show that this geometry and its rank ≥ 3 residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then ...

A Combinatorial Theorem for Trees

Polyunsaturated posets and graphs and the Greene-Kleitman theorem

A partition of a finite poset into chains places a natural upper bound on the size of a union of k antichains. A chain partition is k-saturated if this bound is achieved. Greene and Kleitman [9] proved that, for each k, every finite poset has a simultaneously kand k + 1-saturated chain partition. West [15] showed that the Greene-Kleitman Theorem is best-possible in a strong sense by exhibiting,...