New polynomial and multidimensional extensions of classical partition results

In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.Comment: Some typos, including a terminology confusion involving the words `clique' and `shape', were fixe

Partition regularity and multiplicatively syndetic sets

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green-Tao and Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x, d, x + d, x + 2d\}$.Comment: 29 pages. v3. Referee comments incorporated, accepted for publication in Acta Arithmetic

Partition Regularity of Nonlinear Diophantine Equations

Ramsey Theory and partition regularity problems are interesting settings of combinatorics that investigate structural properties of families of sets. More precisely, a collection of a sets of A, namely F, is partition regular on the set A if, whenever A is finitely partitioned in C_1,...,C_r, then there exists an index j in {1,...,r} and an element of F contained in C_j. Our interest is focused on diophantine equations. In particular we answer to the following question: given a polynomial P, is P partition regular over the natural numbers? This means: given a finite partition (or colouring) of natural numbers, can we find monochromatic solutions of P? The thesis is structured in four chapters. The first chapter lays the foundations of the rest of the thesis. It starts with the theory of ultrafilters which are important and multifaced mathematical objects, whose definition can be formulated in several languages: from set theory, as maximal families of closed under finite intersection sets, to measure theory, as {0,1}-valued finitely additive measures on a given space, to algebra as maximal ideals of ring of function F^I. The chapter continues with a brief dissertation about nonstandard analysis that was created in the early 1960s by the mathematician Abraham Robinson. In particular we focus on hypernatural numbers and their properties to prove the main results of this thesis. We show that the theory of ultrafilters and nonstandard analysis are strictly connected, and they have many applications in other fields of mathematics, as combinatorics or topology. Though this, in the second chapter we can prove some important well-known results that concern partition regularity. We focus our attention on Ramsey Theory, a branch of combinatorics that studies the conditions under which order must appear. Typically, Ramsey problems are connected to questions of the form: how many elements of a given structure should there be to make true a particular property? The begin of this theory is dated back to 1928 when Frank Plumpton Ramsey published his paper "On a problem of formal logic". The paper has led to a large area of combinatorics now known as Ramsey Theory and several important results arose from it in the last century. The most important results that are relevant to our purposes are: Schur's Theorem (1916), Rado's Theorem (1933) that gives a characterisation of the homogeneous systems to which a monochromatic solution can be found in any finite colouring of the natural numbers, Van der Waerden's Theorem (1927), Hindman’s Theorem (1974), and Milliken-Taylor's Theorem (1975). Rado's Theorem completely settled the characterisation of partition regularity of the linear polynomials, and it is the starting point from which the heart of this thesis develops: the partition regularity of nonlinear equations. Actually, the third chapter is dedicated to proving the partition regularity of a few particular equations. Furthermore we give necessary conditions to say when a polynomial is partition regular. These conditions depend on Rado's Theorem and on the degree of the nonlinear variables. In the last fourth chapter we investigate the non-partition regularity of large classes of nonlinear equations. Starting from two simple non partition regular polynomials, x^2+y^2-z and x+y-z^2, we aim at extending these examples. The first step toward the generalisation is to modify the exponents: we prove that the equations x^n+y^n=z^k and x^n+y^m=z^k with n,m,k mutually distinct are non-partition regular. Subsequently we increase the numbers of variables and we prove that also the following equations are non-partition regular: x_1^n+...+x_m^n = y^k with m>1 and kn and there exists a prime p that divides m and p^{k-n} does not divide m; x_1^n+...+x_m^n = y^{n+1}. In the end considering polynomials with coefficients c_j not equal to 1, under suitable conditions on the c_j, we have that the two following equations are non-partition regular: c_1x_1^n+...+c_nx_m^n = y^k, with k1; c_1x_1^n+...+c_nx_m^n = y^{n+1}, with m>1

Counting Configurations in Designs

AbstractGiven a t-(v, k, λ) design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and λ and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design

Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of \nats possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech compactification \beta \nats . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225, arXiv:1301.511

Distinguishing subgroups of the rationals by their Ramsey properties

A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q, and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way. Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q, each of which has a system that is partition regular over it but not over its predecessors. We actually prove something stronger: our main result is that if R and S are subrings of Q with R not contained in S, then there is a system that is partition regular over R but not over S. This implies, for example, that the chain above may be taken to be uncountable.Comment: 14 page

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary domain or a noetherian ring. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.Comment: 19 page