66,038 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
The proof-theoretic strength of Ramsey's theorem for pairs and two colors
Ramsey's theorem for -tuples and -colors () asserts
that every k-coloring of admits an infinite monochromatic
subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and
two colors, namely, the set of its consequences, and show that
is conservative over . This
strengthens the proof of Chong, Slaman and Yang that does not
imply , and shows that is
finitistically reducible, in the sense of Simpson's partial realization of
Hilbert's Program. Moreover, we develop general tools to simplify the proofs of
-conservation theorems.Comment: 32 page
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
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