Lectures on ideal dichotomy
More results in polychromatic Ramsey theory
We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and for every α <ω 2; (2) 2ω = ω 2 and .
Partition properties of ω1 compatible with CH
A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.
Hausdorff ’s theorem for posets that satisfy the finite antichain property
Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC). Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1)...
Lusin sequences under CH and under Martin's Axiom
Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.
Squares with diamonds and Souslin trees with special squares
Event-Based Proof of the Mutual Exclusion Property of Peterson’s Algorithm
Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined...
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