We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degre...

Abadom, P. N. & Scholefield, P. G. (1962). Canad. J. Biochem. Physiol. 40, 1603. Bellamy, D., Phillips, J. G., Jones, C. & Leonard, R. A. (1962). Biochem. J. 85, 537. Bihler, I. & Crane, R. K. (1962). Biochim. biophys. Acta, 59, 78. Birmingham, M. K., Kurlents, E., Lane, R., Muhlstock, B. & Traikov, H. (1960). Canad. J. Biochem. Physiol. 38, 1077. Csaky, T. Z. & Thale, M. (1960). J. Physiol. 15...

Using the monotonicity theorem of L. van den Dries for RCF-definable real functions, and a further result of that author about RCF-definable equivalence relations on R, we show that the theory of order with successors is not interpretable in the theory RCF. This confirms a conjecture by J. Mycielski, P. Pudlák and A. Stern. 1. Let RCF be the theory of real closed fields. We may view RCF as the ...

We show that for an atomless complete Boolean algebra 8 of density < 2N°, the Banach-Mazur, the split and choose, and the Ulam game on S are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff B adds a real; Nonempty wins iff B has a (r-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáá. 0. Introduction. In [Jel an...

We prove that if a graph embeds on a surface with all edges suitably short, then the vertices of the graph can be five-colored. The motivation is that a graph embedded with short edges is locally a planar graph and hence should not require many more than four colors. Introduction. It is well known [4, 15] that a graph embedded on a surface of genus k > 0, the sphere with k handles, can always b...

We show that the Ehrenfeucht-Mycielski sequence U is strongly balanced in the following sense: for any finite word w of length k, the limiting frequency of w in U is 2. 1. The Ehrenfeucht-Mycielski Sequence In [2] Ehrenfeucht and Mycielski introduced an infinite binary word based on avoiding repetitions. More precisely, to construct the Ehrenfeucht-Mycielski (EM) sequence U , start with a singl...

1 I n t r o d u c t i o n Let G = (V, E) be a finite directed graph in which each vertex has at least one edge going out of it. Let w : E , { W , . . . , 0 , . . . , W} be a function that assigns an integral weight to each edge of G. Ehrenfeucht and Mycielski [EM79] studied the following infinite two-person game played on such a graph. The game starts at a vertex a0 E V. The first player choose...

We study how the entropic uncertainty relation for position and momentum conjugate variables is minimized in the subspace of one-dimensional antisymmetric wave functions. Based partially on numerical evidence and partially on analytical results, a conjecture is presented for the sharp bound and for the minimizers. Conjectures are also presented for the corresponding sharp Hausdorff-Young inequa...

A number of recent papers have been devoted to the study of prevalence, a generalization of the property of being of full Haar measure to topological groups which need not have a Haar measure, and the dual concept of shyness. These concepts give a notion of “largeness” which often differs from the category analogue, comeagerness, and may be closer to the intuitive notion of “almost everywhere.”...

We survey determinacy, definability, and complexity issues of Banach-Mazur games on finite and infinite graphs. Infinite games where two players take turns to move a token through a directed graph, thus tracing out an infinite path, have numerous applications in different branches of mathematics and computer science. In the usual format, the possible moves of the players are given by the edges ...