12,178 research outputs found
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
update
A notion of selective ultrafilter corresponding to topological Ramsey spaces
We introduce the relation of "almost-reduction" in an arbitrary topological
Ramsey space R, as a generalization of the relation of "almost-inclusion" on
the space of infinite sets of natural numbers (the Ellentuck space). This leads
us to a type of ultrafilter U on the set of first approximations of the
elements of R which corresponds to the well-known notion of "selective
ultrafilter" on N, the set of natural numbers. The relationship turns out to be
rather exact in the sense that it permits us to lift several well-known facts
about selective ultrafilters on N and the Ellentuck space to the ultrafilter U
and the Ramsey space R. For example, we prove that the Open Coloring Axiom
holds in M[U], where M is a Solovay model. In this way we extend a result due
to Di Prisco and Todorcevic which gives the same conclusion for the Ellentuck
space.Comment: 24 pages; submitted to Mathematical Logic Quarterl
Size-Ramsey numbers of structurally sparse graphs
Size-Ramsey numbers are a central notion in combinatorics and have been
widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and
Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of
-vertex graphs with constant maximum degree . For example, graphs
which also have constant treewidth are known to have linear size-Ramsey
numbers. On the other extreme, the canonical examples of graphs of unbounded
treewidth are the grid graphs, for which the best known bound has only very
recently been improved from to by Conlon, Nenadov and
Truji\'c. In this paper, we prove a common generalization of these results by
establishing new bounds on the size-Ramsey numbers in terms of treewidth (which
may grow as a function of ). As a special case, this yields a bound of
for proper minor-closed classes of graphs. In
particular, this bound applies to planar graphs, addressing a question of Wood.
Our proof combines methods from structural graph theory and classic
Ramsey-theoretic embedding techniques, taking advantage of the product
structure exhibited by graphs with bounded treewidth.Comment: 21 page
On Some Edge Folkman Numbers Small and Large
Edge Folkman numbers Fe(G1, G2; k) can be viewed as a generalization of more commonly studied Ramsey numbers. Fe(G1, G2; k) is defined as the smallest order of any Kk -free graph F such that any red-blue coloring of the edges of F contains either a red G1 or a blue G2. In this note, first we discuss edge Folkman numbers involving graphs Js = Ks − e, including the results Fe(J3, Kn; n + 1) = 2n − 1, Fe(J3, Jn; n) = 2n − 1, and Fe(J3, Jn; n + 1) = 2n − 3. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on Fe(J4, J4; k) for all k \u3e 4
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