On a problem concerning ordered colourings

On the Complexity of Ordered Colourings

A Problem on Ordered Sets

1. Let S be an ordered set, of power 1 SI and order type 8 = 4, We denote by #* the converse of 4, i.e. the order type obtained from + by replacing every order relation x < y by the corresponding relation y < x, and by wrc the least ordinal number of power N,. It is easy to see that, if ISl=tQ, then S contains a subset S’ such that either s’ = w,, or 8’ = w**. For cardinals N, > w’. the corresp...

On a Problem concerning Permutation Polynomials

Let S(f) denote the set of integral ideals / such that / is a permutation polynomial modulo i", where / is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets S which may be written in the form S(f). Introduction. A polynomial f(x) with coefficients in a commutative ring R is said to be a permutation polynomial modulo an ideal I oi R (abb...

On A Problem Concerning Parameter Free Induction

8 0. Preliminaries In this note we comment on the relationships among local versions of the parameter free induction, collection and pigeonhole schemas. We start by defining the versions that will interest us. We work in the usual language of arithmetic to which a new constant symbol a has been added. P-denotes a finite set of axioms such that if M C P-, then M is the nonnegative part of a comm...

On a Problem Concerning the Weight Functions

Let X be a finite set with n elements. A function f : X −→ R such that ∑ x∈X f (x) ≥ 0 is called a n-weight function. In 1988 Manickam and Singhi conjectured that, if d is a positive integer and f is a n-weight function with n ≥ 4d there exist at least (n−1 d−1 ) subsets Y of X with |Y | = d for which ∑ y∈Y f (y) ≥ 0. In this paper we study this conjecture and we show that it is true if f is a ...