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Equidecomposability of Jordan domains under groups of isometries

M. Laczkovich (2003)

Fundamenta Mathematicae

Let G d denote the isometry group of ℝ d . We prove that if G is a paradoxical subgroup of G d then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system ℱ d of Jordan domains with differentiable boundaries and of the same volume such that ℱ d has the cardinality of the continuum, and for every amenable subgroup G of G d , the elements of ℱ d are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

Equitable coloring of Kneser graphs

Robert Fidytek, Hanna Furmańczyk, Paweł Żyliński (2009)

Discussiones Mathematicae Graph Theory

The Kneser graph K(n,k) is the graph whose vertices correspond to k-element subsets of set {1,2,...,n} and two vertices are adjacent if and only if they represent disjoint subsets. In this paper we study the problem of equitable coloring of Kneser graphs, namely, we establish the equitable chromatic number for graphs K(n,2) and K(n,3). In addition, for sufficiently large n, a tight upper bound on equitable chromatic number of graph K(n,k) is given. Finally, the cases of K(2k,k) and K(2k+1,k) are...

Euler’s Partition Theorem

Karol Pąk (2015)

Formalized Mathematics

In this article we prove the Euler’s Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf’s lecture notes [28] (see also [1]). Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [27].

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