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Infinite monochromatic patterns in the integers [article]

Mauro Di Nasso
2022 arXiv   pre-print
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.  ...  The proofs use algebra in the space of ultrafilters βℤ.  ...  Introduction Many of the classic results in arithmetic Ramsey Theory are about the existence of monochromatic patterns found in any given finite coloring of the integers or of the natural numbers.  ... 
arXiv:2105.09541v2 fatcat:2vu7vkd7fjgf3a57tuaeqbrjya

Monochromatic infinite sumsets [article]

Imre Leader, Paul A. Russell
2017 arXiv   pre-print
We show that there is a rational vector space V such that, whenever V is finitely coloured, there is an infinite set X whose sumset X+X is monochromatic.  ...  Our example is the rational vector space of dimension {_0,2^_0,2^2^_0,... }. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below _ω.  ...  Let k be a positive integer, and suppose Q ω is k-coloured. Then there is an infinite set X ⊂ Q ω such that the sumset X + X is monochromatic. Proof. Let c be the given k-colouring of Q ω .  ... 
arXiv:1707.08071v1 fatcat:ekbihpmhinagzg7el3coabvcru

Canonical theorems for colored integers with respect to some linear combinations [article]

Maria Axenovich, David S. Gunderson, Hanno Lefmann
2021 arXiv   pre-print
Hindman proved in 1979 that no matter how natural numbers are colored in r colors, for a fixed positive integer r, there is an infinite subset X of numbers and a color t such that for any finite non-empty  ...  These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.  ...  Indeed, the set Y 3 cannot be monochromatic in color 1 under ∆ 5 , as there is no infinite strictly decreasing sequence of non-negative integers.  ... 
arXiv:2109.10249v1 fatcat:3qpj5efalnf2dgdfuf63tkuv3y

Fermat's Last Theorem Implies Euclid's Infinitude of Primes [article]

Christian Elsholtz
2020 arXiv   pre-print
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of a+b=c implies that there exist infinitely many primes.  ...  In particular, for small exponents such as n=3 or 4 this gives a new proof of Euclid's theorem, as in this case Fermat's last theorem has a proof that does not use the infinitude of primes.  ...  The author would like to thank the referees, the editor, R. Dietmann, J. Erde, I. Leader, R. Meštrović, J.-C. Schlage-Puchta and A. Wiles for useful comments on the manuscript.  ... 
arXiv:2009.06722v2 fatcat:juojyc4enfgtvhagiuetxb2ypm

On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions

Tom C. Brown, Ronald L. Graham, Bruce M. Landman
1999 Canadian mathematical bulletin  
We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer.  ...  Specifically, we want to know for which sets A, of positive integers, the following statement holds: for all positive integers r and k, there exists a positive integer n = w (k, r) such that for every  ...  Acknowledgements The authors are grateful for suggestions from the referee, which led to significant improvements of this paper.  ... 
doi:10.4153/cmb-1999-003-9 fatcat:aqo5fdfxozc5vcvvmimxxbnzre

Monochromatic arithmetic progressions with large differences

Tom C. Brown, Bruce M. Landman
1999 Bulletin of the Australian Mathematical Society  
To simplify the notation in this case, we assume that / is a function from the positive integers to the positive integers. If g is a function, the symbol <7 (r ' will denote the rth iterate of g.  ...  If this sequence does not contain infinitely many 001's (that is g(y) = 0, g(y + 1) = 0, g(y + 2) = 1 for infinitely many y's) or infinitely many 110's, then the sequence has a tail consisting of 000..  ... 
doi:10.1017/s0004972700033293 fatcat:er55btg7qbdrzp7d6ffgulvewq

Exponential Patterns in Arithmetic Ramsey Theory [article]

Julian Sahasrabudhe
2016 arXiv   pre-print
In particular, for every finite colouring of the natural numbers one can find a monochromatic quadruple of the form { a,b,ab,a^b }, where a,b>1.  ...  For example, as a corollary to our main theorem, we show that for every n ∈N and for every finite colouring of the natural numbers, we may find a monochromatic set including the integers x_1,...  ...  In what follows, we drop the set braces around the monochromatic patterns. Infinite linear systems of equations have also been studied.  ... 
arXiv:1607.08396v2 fatcat:ypscklisa5cu5ah5uyzuym644a

Monochromatic Solutions to Systems of Exponential Equations [article]

Julian Sahasrabudhe
2016 arXiv   pre-print
The aim of this paper is to classify precisely which of these systems admit a monochromatic solution (X_i,Y_i =1) in an arbitrary finite colouring of the natural numbers.  ...  We define the exponential system of equations E(R,(C_k(i,j)_i,j,k) to be the system X_i^Y_1^C_1(i,j)... Y_n^C_n(i,j) = X_j , for (i,j) ∈ R , in variables X_1,...,X_n,Y_1,...,Y_n.  ...  for monochromatic patterns in more colours.  ... 
arXiv:1608.00109v1 fatcat:5pmvo66mjzbzhep362nsvetd3u

Monochromatic solutions to x+y=z2 in the interval [N,cN4]

Péter Pál Pach
2018 Bulletin of the London Mathematical Society  
In fact, they showed the existence of a monochromatic solution in every interval [N,cN^8] with large enough N.  ...  Green and Lindqvist proved that for any 2-colouring of N, there are infinitely many monochromatic solutions to x+y=z^2.  ...  The author would like to thank Péter Csikvári and Hong Liu for reading an earlier version of this note and for their useful comments.  ... 
doi:10.1112/blms.12207 fatcat:x4jw7ob6tncbxeemw5pfehgdji

Overcoming the Rayleigh Criterion Limit with Optical Vortices

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, C. Barbieri
2006 Physical Review Letters  
The second source, crossing the fork-hologram in positions different from the optical center, acquires different OAM values and generates non-symmetric L-G patterns.  ...  We experimentally and numerically tested the separability of two independent equally-luminous monochromatic and white light sources at the diffraction limit, using Optical Vortices (OV), related to the  ...  because it acquires a non-integer OAM value producing an asymmetric pattern.  ... 
doi:10.1103/physrevlett.97.163903 pmid:17155396 fatcat:64jxyg2oqvbhxfq2blvlg6vi7i

Page 3728 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
Any (finite or infinite) sequence of integers can obviously be 2- colored so that no finite monochromatic subsequence sums to zero.  ...  In this paper the author introduces an extension of the 3x + 1 function T:Z* — Z* to Z,[/], the 2-adic integers adjoined with /. If [x] denotes the equivalence class of x in Z2[i]/2Z.  ... 

Abelian maximal pattern complexity of words

2013 Ergodic Theory and Dynamical Systems  
In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence.  ...  We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection.  ...  In [3] , the first and third authors introduced a different notion of the complexity of an infinite word called the maximal pattern complexity: For each positive integer k, let Σ k (N) denote the set  ... 
doi:10.1017/etds.2013.51 fatcat:kakjxuapiravjhmu56jlt2go6m

Finding unavoidable colorful patterns in multicolored graphs [article]

Matthew Bowen and Ander Lamaison and Alp Müyesser
2020 arXiv   pre-print
Consider the smallest natural number R_ε^r(H) such that for all n≥ R_ε^r(H), all ε-balanced colorings χ of K_n contain a subgraph isomorphic to H in its coloring.  ...  We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of K_n where each color is well-represented.  ...  Acknowledgements We would like to thank Benny Sudakov for bringing to our attention the results in [3] and [9] .  ... 
arXiv:1807.02780v3 fatcat:d7e35j5livdzlfcks33fdbbk4m

"Weak yet strong" restrictions of Hindman's Finite Sums Theorem

Lorenzo Carlucci
2017 Proceedings of the American Mathematical Society  
Pick n in F S =b (H) so large that the following three points are satisfied:  ...  Consider the following computable coloring of N in 2 colors. c(n) = V SG(n) mod 2.  ...  Acknowledgment I would like to thank the anonymous referee for suggestions that improved the presentation of the paper.  ... 
doi:10.1090/proc/13856 fatcat:sald7425und7fgo3773m6lfxqa

The evolution of unavoidable bi-chromatic patterns and extremal cases of balanceability [article]

Yair Caro, Adriana Hansberg, Amanda Montejano
2022 arXiv   pre-print
In particular, we show that, in 2-colorings whose graphs induced by each of the colors are both free from an induced matching on r edges, the appearance of the unavoidable patterns is already granted with  ...  We study the color patterns that, for n sufficiently large, are unavoidable in 2-colorings of the edges of a complete graph K_n with respect to min{e(R), e(B)}, where e(R) and e(B) are the numbers of red  ...  The third author was partially supported by PAPIIT IN116519, PAPIIT IG100822 and CONACyT project 282280.  ... 
arXiv:2204.04269v1 fatcat:vhin3gldsvb7zcbjobgsesye74
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