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 less than or equal to Z(infinity ) exhibits the Specker phenomenon iff every homomorphism G --> Z maps almost all unit vectors to 0. Let se be the size of the smallest G les s than or equal to Z(infinity) exhibiting the Specker phenomenon. (2) Given an uncountable-dimensional vector space E equipped with a symmet ric bilinear form phi over an at most countable field K, (E, phi) is s trongly Gross iff for all countable-dimensional U less than or equal t o E, we have dim(U-perpendicular to) less than or equal to omega. Blas s showed that the Specker phenomenon is closely related to a combinato rial phenomenon he called evading andpl edicting. We prove several add itional results (both theorems of ZFC and independence proofs) about e vading and predicting as well as se, and relate a Luzin-style property associated with evading to the existence of strong Gross spaces.