Cellularity of first countable spaces (Q1096917)

We find subspaces of the Pixley-Roy space on the irrationals which are (1) a first countable ccc space which does not have a \(\sigma\)-linked base, (2) for each \(n>1\), a first countable space which has a \(\sigma\)-n- linked base but which does not have a \((\sigma -n+1)\)-linked base and (3) a first countable space which has, for each \(n>1\), a \(\sigma\)-n-linked base but which does not have a \(\sigma\)-centered base. It is consistent with \(\neg Ch\) that (1) and (2) have cardinality \(\aleph_ 1\). (3) is constructed from a graph G on the continuum c which is not the union of countably many complete subgraphs but has no uncountable pairwise incompatible family of finite complete subgraphs (complete subgraphs A and B are compatible if there is a complete subgraph C which contains A and B).