Paradoxical decompositions and finitary rules

数理解析研究所講究録, 2005

Electronic Notes in Discrete Mathematics, 2002

We introduce a class of "chromatic" graph parameters that includes the chromatic number, the circular chromatic number, the fractional chromatic number, and an uncountable horde of others. We prove some basic results about this class and pose some problems.

Archive for Mathematical Logic, 2008

We say that a coloring \({c: [\kappa]^n\to 2}\) is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some \({\aleph_1}\) -preserving extension of the set-theoretic universe. Given an arbitrary coloring \({c:[\kappa]^n\to 2}\) , we define a forcing notion \({\mathbb P_c}\) that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that \({\mathbb P_c}\) is c.c.c. if and only if c is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding \({\aleph_1}\) Cohen reals to any model of set theory introduces a coloring \({c:[\aleph_1]^2 \to 2}\) which is potentially continuous but not continuous. \({\aleph_1}\) has no uncountable c-homogeneous subset in the Cohen extension, but such a set can be introduced by forcing. The potential continuity of c can be destroyed by some c.c.c. forcing.

Topological Methods in Nonlinear Analysis, 2015

In some recent papers the classical 'splitting necklace theorem' is linked in an interesting way with a geometric 'pattern avoidance problem', see Alon et al. (Proc. Amer. Math. Soc., 2009), Grytczuk and Lubawski (arXiv:1209.1809 [math.CO]), and Lasoń (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of R d such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lasoń, we show that for every collection µ 1 , ..., µ 2d−1 of 2d − 1 continuous finite measures on R d , there exist two nontrivial axis-aligned d-dimensional cuboids (rectangular parallelepipeds) C 1 and C 2 such that µ i (C 1) = µ i (C 2) for each i ∈ {1, ..., 2d − 1}. We also show by examples that the bound 2d − 1 cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.

Journal of Graph Theory, 2007

In an ordinary list multicoloring of a graph, the vertices are "colored" with subsets of pre-assigned finite sets (called "lists") in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel.

Journal of Mathematical Psychology, 1989

,} be a finite set and Q be an algebra of subsets of X called events. Let 2 be a qualitative probability relation on Q x Q. A probability measure p is said to uniquely agree with 2 if, for all A and B in 52, A 2 B if and only if p(A) > p(B), and p is the only probability measure with that property. We give a sufficient condition for the existence of a uniquely agreeing probability measure which is signiticantly simpler and more general than Lute's (1967, Annals of Mathematical Stutistics, 38, 780-786) condition.

Information Sciences, 2013

We explore the relationship between possibility measures (supremum preserving normed measures) and p-boxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating that only those p-boxes who have 0-1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes. Doing so, we arrive at a new rule of combination for possibility measures, for the independent case.

1998

We introduce resource-bounded betting games, and propose a generalization of Lutz's resource-bounded measure in which the choice of next string to bet on is fully adaptive. Lutz's martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong pseudo-random number generators exist, then betting games are equivalent to martingales, for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz measures are important open problems: the class of polynomial-time Turing-complete languages in EXP, and its superclass of polynomial-time Turing-autoreducible languages. If an EXP-martingale succeeds on either of these classes, or if betting games have the nite union property possessed by Lutz's measure, one obtains the non-relativizable consequence BPP 6 = EXP. We also show that if EXP 6 = MA, then the polynomial-time truth-table-autoreducible languages have Lutz measure zero, whereas if EXP = BPP, they have measure one.

The Review of Symbolic Logic

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

Proceedings of the American Mathematical Society, 2019

A particular case of the Hindman–Galvin–Glazer theorem states that, for every partition of an infinite abelian group G G into two cells, there will be an infinite X ⊆ G X\subseteq G such that the set of its finite sums { x 1 + ⋯ + x n ∣ n ∈ N ∧ x 1 , … , x n ∈ X are distinct } \{x_1+\cdots +x_n \mid n\in \mathbb N\wedge x_1,\ldots ,x_n\in X\text { are distinct}\} is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) X X . On the other hand, a recent result of Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form F S ( X ) \mathrm {FS}(X) , for X X of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komjáth’s result, and we show that, in a sense, this generalization is the strongest possible.