81 research outputs found
On constructions with -cardinals
We propose developing the theory of consequences of morasses relevant in
mathematical applications in the language alternative to the usual one,
replacing commonly used structures by families of sets originating with
Velleman's neat simplified morasses called -cardinals. The theory of related
trees, gaps, colorings of pairs and forcing notions is reformulated and
sketched from a unifying point of view with the focus on the applicability to
constructions of mathematical structures like Boolean algebras, Banach spaces
or compact spaces.
A new result which we obtain as a side product is the consistency of the
existence of a function
with the
appropriate -version of property for regular
satisfying .Comment: Minor correction
Cardinals Beyond Choice and the HOD-Dichotomy
Treballs Finals del MĂ ster de LĂČgica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 2019-2020. Tutor: Joan BagariaIn the 2019 paper "Large Cardinals Beyond Choice" [1], Bagaria, Koellner and Woodin apply the
large cardinal techniques and results developed fromWoodin's work on the HOD-Dichotomy to determine
the structural resemblance of HOD to V . Whereas standard inner model theory attempts to nd suitable
inner models for large cardinals, this new program is aimed at exploring very large cardinals that \break"
the resemblance of HOD to V . This paper attempts to explain in full detail the tools and arguments
required for that body of work
Computing the Homology of Semialgebraic Sets. II: General formulas
We describe and analyze a numerical algorithm for computing the homology
(Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean
formulas. The algorithm works in weak exponential time. This means that outside
a subset of data having exponentially small measure, the cost of the algorithm
is single exponential in the size of the data. This extends the previous work
of the authors in arXiv:1807.06435 to arbitrary semialgebraic sets.
All previous algorithms proposed for this problem have doubly exponential
complexity.Comment: 33 pages, 4 figure
Ramsey properties of algebraic graphs and hypergraphs
One of the central questions in Ramsey theory asks how small can be the size
of the largest clique and independent set in a graph on vertices. By the
celebrated result of Erd\H{o}s from 1947, the random graph on vertices with
edge probability , contains no clique or independent set larger than
, with high probability. Finding explicit constructions of graphs
with similar Ramsey-type properties is a famous open problem. A natural
approach is to construct such graphs using algebraic tools. Say that an
-uniform hypergraph is \emph{algebraic of complexity
} if the vertices of are elements of
for some field , and there exist polynomials
of degree at most
such that the edges of are determined by the zero-patterns of
. The aim of this paper is to show that if an algebraic graph
(or hypergraph) of complexity has good Ramsey properties, then at
least one of the parameters must be large. In 2001, R\'onyai, Babai and
Ganapathy considered the bipartite variant of the Ramsey problem and proved
that if is an algebraic graph of complexity on vertices, then
either or its complement contains a complete balanced bipartite graph of
size . We extend this result by showing that such
contains either a clique or an independent set of size
and prove similar results for algebraic hypergraphs of constant complexity. We
also obtain a polynomial regularity lemma for -uniform algebraic hypergraphs
that are defined by a single polynomial, that might be of independent interest.
Our proofs combine algebraic, geometric and combinatorial tools.Comment: 23 page
On irresponsible homomorphisms and strong duality
This thesis looks at algebras with positive primitively defined binary relations that are almost re- flexive, anti-symmetric, and transitive and provides new machinery for determining when these algebras are not strongly dualizable.algebrabinaryanti-symmetricdualize
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