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Evasion and prediction --- the Specker phenomenon and Gross spaces

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.