This is a study of several notions of size of subsets of groups. The first part (Sections 3–5) concerns a purely algebraic setting with the underlying group discrete. The various notions of size considered there are similar to each other in that each of them assesses the size of a set using a family of measures and translations of the set; they differ in the type of measures used and the type o...

The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence G n of triangle-free graphs with (G n ) = n. In this note, we calculate the fractional chromatic number of G n and show that this sequence of numbers satis es the unexpected recurrence a n+1 = a n + 1 a n .

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G)+1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chr...

An R~-universally extending ordered field of power N, is constructed for each regular power N,, where 0 < of -< On and Zt~<~ 2• <-X,. When ,R, is inaccessible, the structure is either a (set) model of J. H. Conway's ordered field No or an isomorphic copy of No depending on whether or not S~. is a set or a proper class. An early version of this paper was delivered to the Joint Colloquium on Hist...

Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that (V α ,∈) satisfies φ(x) ∧ ¬φ(y) . We examine the relationship between LM and some other axioms of ...

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the hom...

With each of the classical tree–like forcings adjoining a new real, one can associate a σ– ideal on the reals in a natural way. For example, the ideal of Marczewski null sets s corresponds to Sacks forcing S, while the ideal of nowhere Ramsey sets r corresponds to Mathias forcing R. We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal P2...

Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure μ on G such that μ(B+g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if G...

Let H be a bipartite graph and let Gn be the Mycielski graph with χ(G) = n, n ≥ 4. Then the chromatic number of the strong product of Gn by H is at most 2n− 2. We use this result to show that there exist strong products of graphs in which a projection of a retract onto a factor is not a retract of the factor. We also show that in the Cartesian product of graphs G and H, any triangles of G trans...

We prove that the chromatic Ramsey number of every odd wheel W2k+1, k ≥ 2 is 14. That is, for every odd wheel W2k+1, there exists a 14-chromatic graph F such that when the edges of F are two-coloured, there is a monochromatic copy of W2k+1 in F , and no graph F with chromatic number 13 has the same property. We ask whether a natural extension of odd wheels to the family of generalized Mycielski...