  We study the set--theoretic combinatorics underlying the following two
algebraic phenomena.
  (1) A subgroup G leq Z^omega exhibits the Specker phenomenon iff every
homomorphism G to Z maps almost all unit vectors to 0. Let se be the size of
the smallest G leq Z^omega exhibiting the Specker phenomenon.
  (2) Given an uncountably dimensional vector space E equipped with a symmetric
bilinear form Phi over an at most countable field KK, (E,Phi) is strongly Gross
iff for all countably dimensional U leq E, we have dim(U^perp) leq omega.
  Blass showed that the Specker phenomenon is closely related to a
combinatorial phenomenon he called evading and predicting. We prove several
additional results (both theorems of ZFC and independence proofs) about evading
and predicting as well as se, and relate a Luzin--style property associated
with evading to the existence of strong Gross spaces.
