A Kruskal–Katona type theorem for graphs

AbstractA bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal–Katona theorem. A bound on non-consecutive clique numbers is also proven

Obtainable Sizes of Topologies on Finite Sets

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies.Comment: Final version, to appear in Journal of Combinatorial Theory, Series