This is an announcement for the paper "Weakly null sequences in the Banach space C(K)" by I. Gasparis, E. Odell, and B. Wahl. Abstract: The hierarchy of the block bases of transfinite normalized averages of a normalized Schauder basic sequence is introduced and a criterion is given for a normalized weakly null sequence in C(K), the Banach space of scalar valued functions continuous on the compact metric space K, to admit a block basis of normalized averages equivalent to the unit vector basis of c_0, the Banach space of null scalar sequences. As an application of this criterion, it is shown that every normalized weakly null sequence in C(K), for countable K, admits a block basis of normalized averages equivalent to the unit vector basis of c_0. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03 Remarks: 36 pages The source file(s), gow5.tex: 137843 bytes, is(are) stored in gzipped form as 0402202.gz with size 33kb. The corresponding postcript file has gzipped size 147kb. Submitted from: combs@mail.ma.utexas.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0402202 or http://arXiv.org/abs/math.FA/0402202 or by email in unzipped form by transmitting an empty message with subject line uget 0402202 or in gzipped form by using subject line get 0402202 to: math@arXiv.org.