[FOM] Help please on Friedman post.

[Bill Taylor]

In a recent posting by Harvey Friedman, the following results were mentioned:


A. In sufficiently long finite sequences from a finite set,
   certain blocks are subsequences of certain later blocks.

B. In sufficiently tall finite trees, certain initial segments
   are embedded in certain taller initial segments.

C. Multivariate functions on Z have nonsurjective infinite restrictions.

D. Any two countable sets of reals are pointwise continuously comparable.


Can someone please help me out understanding these.  Specifically...


In D, what does "pointwise continuously comparable" mean?

In C, are the restrictions "proper" ones in the sense that both
the domain and the range are the same subset of Z, (allowing that
the domain is actually the nth power of that, of course).

In B, are *rooted* trees being referred to?  ("Tall" suggests so.)
If so, what type of embeddings are being considered, as subtrees or
as homeomorphic to subtrees?

In A, does subsequence mean sub-block or just subset with the induced ordering?

In A and B, can anyone provide an exact figure on
the "sufficiently" and the "certain"?


Thanks in advance.

Bill Taylor