An odd square as a sum of an odd number of odd squares

Disjoint odd integer subsets having a constant odd sum

We prove that for positive k, n and m, the set { 1,3, . ,2n 1) of odd integers contains k disjoint subsets having a constant odd sum m if and only if 9(k-1)Cm <2n-1, or 9k<m<n2/k and n2-mk#2.

An odd characterization of the generalized odd graphs

Article history: Received 27 April 2010 Available online 17 March 2011

Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles

Monsky’s theorem from 1970 states that a square cannot be dissected into an odd number n of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky’s theorem to “constrained framed maps”; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for th...

Modularity of an Odd Icosahedral

Odd Independent Transversals are Odd

We put the final piece into a puzzle first introduced by Bollobás, Erdős and Szemerédi in 1975. For arbitrary positive integers n and r we determine the largest integer ∆ = ∆(r, n), for which any r-partite graph with partite sets of size n and of maximum degree less than ∆ has an independent transversal. This value was known for all even r. Here we determine the value for odd r and find that ∆(...