913 research outputs found
Difference Ramsey Numbers and Issai Numbers
We present a recursive algorithm for finding good lower bounds for the
classical Ramsey numbers. Using notions from this algorithm we then give some
results for generalized Schur numbers, which we call Issai numbers.Comment: 10 page
Partnerships
V diplomskem delu podrobneje obravnavamo Schurov izrek o vsot-prostih particijah in definiramo n-to Schurovo število S(n) kot največje naravno število, za katerega obstaja razbitje množice {1,...,S(n)} na n disjunktnih vsot-prostih podmnožic. Zapišemo prvih nekaj znanih Schurovih števil in določimo meje, znotraj katerih se gibljejo vrednosti večjih, še neznanih Schurovih števil. Omenimo šibka Schurova števila. Schurov izrek formuliramo tudi kot problem barvanja in posledico Ramseyjeve teorije. Za konec si pogledamo, kako je Schurov izrek povezan z zadnjim Fermatovim izrekom. Pokažemo, na kakšen način je Schur poenostavil Dicksonovo trditev, da ima enakost x^n+y^n=z^n pri danem naravnem številu n > 2 netrivialne rešitve v Z_p za vsa dovolj velika praštevila p.In the thesis, Schur\u27s theorem on sum-free partitions is proven and Schur number S(n) is defined as the largest positive integer with the property that the set {1,...,S(n)} can be partitioned into n sum-free subsets. Values of known Schur numbers S(1) to S(5) are given as well as some upper and lower bounds for general S(n). Weak Schur numbers are also defined. Moreover, Schur\u27s theorem is formulated as a graph coloring problem and presented as a corollary of Ramsey theorem. In conclusion, Schur\u27s theorem is linked to Fermat\u27s last theorem. Schur\u27s simplification of Dickson\u27s proof that equation x^n+y^n=z^n for fixed n > 2 has nontrivial solutions in Z_p for all sufficiently large prime p is given
Semi-algebraic Ramsey numbers
Given a finite point set , a -ary semi-algebraic
relation on is the set of -tuples of points in , which is
determined by a finite number of polynomial equations and inequalities in
real variables. The description complexity of such a relation is at most if
the number of polynomials and their degrees are all bounded by . The Ramsey
number is the minimum such that any -element point set
in equipped with a -ary semi-algebraic relation , such
that has complexity at most , contains members such that every
-tuple induced by them is in , or members such that every -tuple
induced by them is not in .
We give a new upper bound for for and fixed.
In particular, we show that for fixed integers , establishing a subexponential upper bound on .
This improves the previous bound of due to Conlon, Fox, Pach,
Sudakov, and Suk, where is a very large constant depending on and
. As an application, we give new estimates for a recently studied
Ramsey-type problem on hyperplane arrangements in . We also study
multi-color Ramsey numbers for triangles in our semi-algebraic setting,
achieving some partial results
Polynomial Schur's theorem
We resolve the Ramsey problem for for all polynomials
over .Comment: 21 page
Superfilters, Ramsey theory, and van der Waerden's Theorem
Superfilters are generalized ultrafilters, which capture the underlying
concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We
establish several properties of superfilters, which generalize both Ramsey's
Theorem and its variant for ultrafilters on the natural numbers. We use them to
confirm a conjecture of Ko\v{c}inac and Di Maio, which is a generalization of a
Ramsey theoretic result of Scheepers, concerning selections from open covers.
Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous
generalization of the theorems of Ramsey, van der Waerden, Schur,
Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much
smaller than the full set of natural numbers.Comment: Among other things, the results of this paper imply (using its
one-dimensional version) a higher-dimensional version of the Green-Tao
Theorem on arithmetic progressions in the primes. The bibliography is now
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