10 papers · 1 filter
A simultaneous extension of Ramsey, Hindman, and Hales-Jewett Theorems
Mauro Di Nasso, Renling Jin
We prove a multidimensional extension of a strong Hales-Jewett theorem that simultaneously and "directly" extends Ramsey's theorem and Hindman's theorem. The proofs show the effect…
A Van der Waerden-free proof of Rado's theorem
Mauro Di Nasso, Lorenzo Luperi Baglini
We present a proof of the sufficiency of Rado's condition for the partition regularity of linear Diophantine equations that avoids any use of van der Waerden's theorem. The proof i…
The magic of tensor products of ultrafilters
Mauro Di Nasso
Tensor products of ultrafilters have special combinatorial features closely related to Ramsey's Theorem, making them useful tools in applications. Here we first review their fundam…
Monochromatic exponential triples: an ultrafilter proof
Mauro Di Nasso, Mariaclara Ragosta
We present a short ultrafilter proof of the existence of monochromatic exponential triples in any finite coloring of the natural numbers. The proof is given from sc…
Infinite monochromatic patterns in the integers
Mauro Di Nasso
We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials. The simplest example is the following. For ev…
Translation invariant filters and van der Waerden's Theorem
Mauro Di Nasso
We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple indu…
Ramsey properties of nonlinear Diophantine equations
Mauro Di Nasso, Lorenzo Luperi Baglini
We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of non…
Intersections of shifted sets
Mauro Di Nasso
We consider shifts of a set by elements from another set , and prove intersection properties according to the relative asymptotic size…
Intersections of sets of distances
Mauro Di Nasso
We isolate conditions on the relative size of sets of natural numbers that guarantee a nonempty intersection of the corresponding sets of distances…
An Elementary Proof of Jin's Theorem with a Bound
Mauro Di Nasso
We present a short proof of Jin's theorem which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or o…