J. Moore proved the consistency of five element basis. In this talk, we will prove the consistency of $3+2^n$ element basis for $n > 1$.

We will use the following concept introduced by S. Todorcevic. For a special coherent tree $T$, $I(T)$ is the ideal consists of subsets of $\omega_1$ such that on which, some uncountable subset never splits.

For some collection $\overset{\rightharpoonup}{\Gamma}$ of pairwise $I(T)$-disjoint $I(T)$-positive sets. $PFA(\overset{\rightharpoonup}{\Gamma})$ is the assertion that for any proper forcing that does not destroy $I(T)$-positivity of any set in $\overset{\rightharpoonup}{\Gamma}$, for any $\omega_1$ collection of dense sets, there is a filter meets them all.

We will first prove the consistency of $PFA(\overset{\rightharpoonup}{\Gamma})$ from a supercompact cardinal. Then prove that $3+2^n$ element basis follows from $PFA(\overset{\rightharpoonup}{\Gamma})$ where $n$ is the size of $\overset{\rightharpoonup}{\Gamma}$.

If we have only one Aronszajn tree in the final model, then $2^n$ is the best we can do for Countryman basis. To get more posibilities, we may want to construct a model with more than one (probably finite) trees. Basis for Aronszajn trees will also be discussed.

This is a joint work with L. Wu.