95 research outputs found

    Infinite Monochromatic Sumsets for Colourings of the Reals.

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    N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of R so that no infinite sumset X + X is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any finite colouring c of R there is an infinite X ⊆ R so that c ↾ X + X is constant

    The number of subsets of integers with no kk-term arithmetic progression

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    Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely many values of nn, the number of subsets of {1,2,,n}\{1,2,\ldots, n\} that do not contain a kk-term arithmetic progression is at most 2O(rk(n))2^{O(r_k(n))}, where rk(n)r_k(n) is the maximum cardinality of a subset of {1,2,,n}\{1,2,\ldots, n\} without a kk-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of nn, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on rk(n)r_k(n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(rk(n))\Theta(r_k(n)) contain superlinearly many kk-term arithmetic progressions. For integers rr and kk, Erd\Ho s asked whether there is a set of integers SS with no (k+1)(k+1)-term arithmetic progression, but such that any rr-coloring of SS yields a monochromatic kk-term arithmetic progression. Ne\v{s}et\v{r}il and R\"odl, and independently Spencer, answered this question affirmatively. We show the following density version: for every k3k\ge 3 and δ>0\delta>0, there exists a reasonably dense subset of primes SS with no (k+1)(k+1)-term arithmetic progression, yet every USU\subseteq S of size UδS|U|\ge\delta|S| contains a kk-term arithmetic progression. Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea.Comment: To appear in International Mathematics Research Notices. This is a longer version than the journal version, containing two additional minor applications of the container metho

    Enumeration of three term arithmetic progressions in fixed density sets

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    Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3: Incorporated feedbac

    Problems on infinite sumset configurations in the integers and beyond

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    In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density.Comment: 37 page

    Partition regularity of a system of De and Hindman

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    We prove that a certain matrix, which is not image partition regular over R near zero, is image partition regular over N. This answers a question of De and Hindman.Comment: 7 page

    An improved lower bound for Folkman's theorem

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    Folkman's Theorem asserts that for each kNk \in \mathbb{N}, there exists a natural number n=F(k)n = F(k) such that whenever the elements of [n][n] are two-coloured, there exists a set A[n]A \subset [n] of size kk with the property that all the sums of the form xBx\sum_{x \in B} x, where BB is a nonempty subset of AA, are contained in [n][n] and have the same colour. In 1989, Erd\H{o}s and Spencer showed that F(k)2ck2/logkF(k) \ge 2^{ck^2/ \log k}, where c>0c >0 is an absolute constant; here, we improve this bound significantly by showing that F(k)22k1/kF(k) \ge 2^{2^{k-1}/k} for all kNk\in \mathbb{N}.Comment: 5 pages, Bulletin of the LM
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