We disprove the 1980 conjecture of János Pach about the cover-decomposability of open convex sets by showing that the unit disc is not cover-decomposable. In fact, our proof easily generalizes to any set with a smooth boundary. We also show that (the suitable variant of) the conjecture holds for unbounded sets. Let C be a collection of sets in R2. We say that C is an m-fold covering if every po...

A well-known theorem of Fong states that over large enough fields of any characteristic, the principal indecomposable modules of a soluble finite group are induced from subgroups of order prime to the characteristic. It is shown that this property in fact characterises soluble finite groups.

Combinatorial functions are used to replace indecomposable cardinals in certain types of set theoretic arguments. This allows us to extend decidability results from nonchoice set theories to set theories with a principle of linear ordering.

In this note, we describe an extension of Hubert's Theorem 90 for K2 of fields to the relative K<i of semilocal PID's containing a field. Most of the results for K2 of fields proven in [M-S and S] then carry over to the relative K% of semilocal PID's containing a field, e.g. computation of the torsion subgroup, and the norm residue isomorphism. Applying this to the semilocal ring of {0,1} in A^...

Let (R,m) be a local ring (commutative and Noetherian). If R is complete (or, more generally, Henselian), one has the Krull-Schmidt uniqueness theorem for direct sums of indecomposable finitely generated R-modules. By passing to the m-adic completion b R, we can get a measure of how badly the Krull-Schmidt theorem can fail for a more general local ring. We assign to each finitely generated R-mo...

In the first two volumes of Fundamenta Mathematica, Knaster and Kuratowski raised the following two questions [15], [16]: (1) If a nondegenerate, bounded plane continuum is homogeneous, is it necessarily a simple closed curve? (2) Does there exist a continuum each subcontinuum of which is indecomposable? Although Knaster settled the second question in 1922 [14], it was to remain until 1948 for ...

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module S has non-trivial cyclic vertex if and only if λ has p-weight 1. 2. If p does not divide n and S(n−r,1 ) is indecomposable then its vertex is a p-Sylow subgroup of Sn−r−1 × Sr. Mathematics Subject Classification...

In 2000, Rees and Shalaby constructed simple indecomposable two-fold cyclic triplesystems for all v ≡ 0, 1, 3, 4, 7, and 9 (mod 12) where v = 4 or v ≥ 12, using Skolem-type sequences.We construct, using Skolem-type sequences, three-fold triple systems having theproperties of being cyclic, simple, and indecomposable for all admissible orders v, withsome possible exception...

In this paper we study the modular structure of the permutation module H ) of the symmetric group S2n acting on set partitions of a set of size 2n into n sets each of size 2, defined over a field of odd characteristic p. In particular we characterize the vertices of the indecomposable summands of H ) and fully describe all of its indecomposable summands that lie in blocks of p-weight at most tw...

A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein’s criterion. Polynomials from these ...