We study some set-theoretic properties of Schmidt’s σ-ideal on R, emphasizing its analogies and dissimilarities with both the classical σ-ideals on R of Lebesgue measure zero sets and of Baire first category sets. We highlight the strict analogy between Schmidt’s ideal on R and Mycielski’s ideal on 2N.

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex of G sees all k colors on its closed neighborhood. We denote Fall(G) the set of all positive integers k for which G has a fall k-coloring. In this paper, we study fall colorings of lexicographic product of graphs and categorical product of graphs and answer a question of [3] about fall colorings of categorical prod...

In this work we give a new lower bound on the chromatic number of a Mycielski graph Mi. The result exploits a mapping between the coloring problem and a multiprocessor task scheduling problem. The tightness of the bound is proved for i = 1; : : : ; 8. c © 2001 Elsevier Science B.V. All rights reserved.

In this paper graphs with uncountable chromatic numbers will be studied. As usual, a graph is an ordered pair G = ( V , E), where V is an arbitrary set (the set of vertices), E is a set of unordered pairs from V (the set of edges). A function : V + x (n a cardinal), is a good coloring of G if and only if f ( x ) ¢ f ( y ) whenever x and y are jo ined i.e. joined vertices get different colors. T...

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of simila...

Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for the descriptive purposes of biological and chemical properties, such as melting and boiling points, toxicity and resistance. Here we consider the following four established irregularity measures: the irregularity index by Albertson, the total irregularity, the variance of vertex degrees and the Collat...

We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of “topo...

A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the LeibnizMycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF , T has a Leibnizian model iff T proves LM. Here we prove: Theorem A. Every complete theory T extending ZF + LM has 2 א 0 nonisomorphic countable Leibnizian models. Theorem B. ...