Abstract
A variety of extremal problems are considered for the members of two particular families of simple 2-complexes. We let Bk denote the complex consisting of k 2-cells all sharing one single edge in common and Wk the complex consisting of k 2-cells sharing exactly one vertex in common. The problems treated here include the following questions for the Bk and the corresponding questions for the Wk.Suppose that for each 2-coloring of the 2-cells of a complex G, either one color contains a monochromatic copy of Bk or the other contains a monochromatic copy of Bℓ. How many 2-cells must G have? How many vertices? How many vertices and how many 2-cells must G have if one of Bk and Bℓ must be "faithfully imbedded" in one of the colors? How do these numbers compare with the "generalized Ramsey numbers" for the pairs (Bk, Bℓ)? What is the smallest integer f(n, k) such that each 2-complex on n vertices having f(n, k) 2-cells contains a copy of Bk?
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