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We provide a combinatorial description of all the minors of the edge version of the Laplacian matrix of a mixed tree. The description involves the common SDR's for the forests obtained by deleting from the tree the edge sets corresponding to the row and column indices of the minor.

The double star S(n, m), where n⩾ m⩾ 0, is the graph consisting of the union of two stars K1,nand K1,mtogether with a line joining their centers. Its ramsey number r(S(n, m)) is the least number psuch that there is a monochromatic copy of S(n, m) in any 2-coloring of the edges of Kp. It is shown that r(S(n, m)) = max(2n+ 1, n+ 2m+ 2) if nis odd and m⩽2 and r(S(n, m)) = max(2n+ 2, n+ 2m+ 2) otherwise, for n⩽ √2morn⩾ 3m.

The category of towers of spaces, , viewed as pro-spaces, appears to be useful in the study of the relation between homology and homotopy of nonsimply connected spaces. We show that this category admits the structure of a closed model category, in the sense of Quillen; notions of fibration, cofibration, and weak equivalence are defined and shown to satisfy fundamental properties that the corresponding notions satisfy in the category of spaces. This enables one to develop a ``homotopy theory'' for pro-spaces.

An example is constructed of a fibration over the torus which induces isomorphisms on all homology groups but which has a nontrivial Serre spectral sequence.