Filters








2,115 Hits in 2.6 sec

A density statement generalizing Schur's theorem

Vitaly Bergelson
1986 Journal of combinatorial theory. Series A  
The straightforward density version of this result is obviously wrong (take odd numbers!). Thus the following problem seems to be natural: to find a density result which generalizes Schur's theorem.  ...  INTRODUCTION For all Ramsey theorems, one can express (but not always prove) the corresponding density statements... R. L. Graham, B. L. Rothschild, J. H.  ...  Part of this work was done while I was a Research Fellow at Imperial College. London. I take this opportunity to thank this college for their hospitality and to thank S.E.R.C. for the fellowship.  ... 
doi:10.1016/0097-3165(86)90074-9 fatcat:3nayp2sfgvb3xm6daygbhsw7pq

Density versions of two generalizations of Schur's theorem

Vitaly Bergelson, Neil Hindman
1988 Journal of combinatorial theory. Series A  
Given a finite partition of the natural numbers, we show that there is one cell with the property that many (in a density sense made precise later) sequences have all of their finite sums in this one cell  ...  Brauer's Theorem [4] simultaneously generalizes Schur's Theorem and van der Waerden's Theorem.  ...  Schur's Theorem has been generalized in several directions. We are concerned here with two of these, the Finite Sum Theorem and Brauer's Theorem.  ... 
doi:10.1016/0097-3165(88)90072-6 fatcat:bgs6bmze3raj5m7vrjbb27qtx4

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.  ...  Schlage-Puchta and A. Wiles for useful comments on the manuscript. The author was partially supported by the Austrian Science Fund (FWF): W1230 and I 4945-N.  ... 
arXiv:2009.06722v2 fatcat:juojyc4enfgtvhagiuetxb2ypm

Page 2 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
Then HM —cq R(x) / TF (x) = Sta)- This result follows from a still more general theorem that is too complicated to be stated here.  ...  Assuming the generalized Riemann hypothesis, the authors investigate the conditions under which the set of points where the normalized remainder term is bigger than y (say) has a (suitably defined) density  ... 

Splitting of integer polynomials over fields of prime order [article]

Shubham Saha
2018 arXiv   pre-print
It is well known that a polynomial ϕ(X)∈Z[X] of given degree d factors into at most d factors in F_p for any prime p.  ...  root in F_q for all sufficiently large primes q, where P∈Z[X] is any polynomial such that P has a root β∈C for which Q(β) is the splitting field of ϕ over Q.  ...  and F has a root in F p iff φ splits in F p .  ... 
arXiv:1802.10562v2 fatcat:dxqbzw4pnzdopjo2q2b4dh7r4m

The Ramsey-type version of a problem of Pomerance and Schinzel

Péter Pál Pach
2012 Acta Arithmetica  
The author is greateful to A. Sárközy for turning his attention to this topic and for the useful hints.  ...  By Schur's theorem the equation x + y = z has a monochromatic solution in N.  ...  It is a consequence of Schur's theorem [9] that Sárközy's original problem always has a solution among the powers of 2. Proposition 1.  ... 
doi:10.4064/aa156-1-1 fatcat:5rtadv3n4fhvngpva3sifkqlgq

Curves of Finite Total Curvature [article]

John M Sullivan
2007 arXiv   pre-print
To explore these ideas, we consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.  ...  This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry  ...  Schur's comparison theorem Schur's comparison theorem [Sch21] is a well-known result saying that straightening an arc will increase the distance between its endpoints.  ... 
arXiv:math/0606007v2 fatcat:cnh7g6rtyvg5td2zymemeekeji

Remarks on the quantum de Finetti theorem for bosonic systems [article]

Mathieu Lewin
2014 arXiv   pre-print
The quantum de Finetti theorem asserts that the k-body density matrices of a N-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed.  ...  In this note we review a construction due to Christandl, Mitchison, K\"onig and Renner valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem.  ...  This yields our Theorem 2.2, an explicit expression of the density matrices of the stateΓ N as a function of the density matrices of the original state Γ N .  ... 
arXiv:1310.2200v3 fatcat:pkmaxhqpmbcuzeugfozkcdpif4

SCHUR'S COLOURING THEOREM FOR NONCOMMUTING PAIRS

TOM SANDERS
2019 Bulletin of the Australian Mathematical Society  
to contain a monochromatic quadruple $(x,y,xy,yx)$ with $xy\neq yx$ .  ...  For $G$ a finite non-Abelian group we write $c(G)$ for the probability that two randomly chosen elements commute and $k(G)$ for the largest integer such that any $k(G)$ -colouring of $G$ is guaranteed  ...  For general finite groups there can be no density result; we refer the reader to the discussion after [BT14, Theorem 11] for more details.  ... 
doi:10.1017/s0004972719000406 fatcat:43p66nz44fcwvhj66tunlzwam4

Solvability of Rado systems in D-sets [article]

Mathias Beiglböck, Vitaly Bergelson, Tomasz Downarowicz, Alexander Fish
2008 arXiv   pre-print
(Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of D-sets.  ...  Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations.  ...  We want to mention a corollary 2 of Rado's Theorem extending Schur's Theorem.  ... 
arXiv:0809.2281v1 fatcat:yt7waoozdrevpa6sa2zmjzcwcy

Remarks on the Quantum de Finetti Theorem for Bosonic Systems

M. Lewin, P. T. Nam, N. Rougerie
2014 Applied Mathematics Research eXpress  
The quantum de Finetti theorem asserts that the k-body density matrices of a N -body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed.  ...  In this note we review a construction due to Christandl, Mitchison, König and Renner [8] valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem.  ...  This yields our Theorem 2.2, an explicit expression of the density matrices of the stateΓ N as a function of the density matrices of the original state Γ N .  ... 
doi:10.1093/amrx/abu006 fatcat:tuedebblerey7cpbncdslc3ljy

Page 1841 of Mathematical Reviews Vol. , Issue 91D [page]

1991 Mathematical Reviews  
The au- thors prove similar density versions of several theorems including Ramsey’s theorem, the vector space Ramsey theorem [R. Graham, K. Leeb and B. L. Rothschild, Adv.  ...  A 43 (1986), no. 2, 338-343; MR 87k:05014] proved the following density version of Schur’s theo- rem: If N=, Cj and e > 0 then there exists i € {1,2,---,m} such that d({n € C;: d(C; C; —n) > d(C;)?  ... 

Solvability of Rado systems in D-sets

M. Beiglböck, V. Bergelson, T. Downarowicz, A. Fish
2009 Topology and its Applications  
(Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of D-sets.  ...  Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations.  ...  ∪I l−1 c l−1 j a ij . We want to mention a corollary 2 of Rado's Theorem extending Schur's Theorem.  ... 
doi:10.1016/j.topol.2009.04.019 fatcat:3l6fsc4thjbbrgbczlpgfsbsty

Extremal results for random discrete structures

Mathias Schacht
2016 Annals of Mathematics  
In particular, we verify a conjecture of Kohayakawa, \L uczak, and R\"odl for Tur\'an-type problems in random graphs. Similar results were obtained by Conlon and Gowers.  ...  We determine the threshold for Szemer\'edi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Tur\'an-type problems  ...  It follows from the 0-statement of Theorem 1.1 in [36] that for any irredundant, density regular, × k integer matrix A of rank and every 1/2 > ε > 0 there exist a c > 0 such that for every sequence of  ... 
doi:10.4007/annals.2016.184.2.1 fatcat:h37q3qgncrbepotos7k36yo3ha

Page 1361 of Mathematical Reviews Vol. , Issue 83d [page]

1983 Mathematical Reviews  
A density version of a geometric Ramsey theorem. J. Combin. Theory Ser. A 32 (1982), no. 1, 20-34. A fundamental result in combinatorics, due to A. W. Hales and R. I. Jewett [Trans. Amer. Math.  ...  The technique used to interpret the analytic identity involves ‘ underlying partitions’ as used in the combinatorial proof of Schur’s Theorem [the author, Proc. Amer. Math.  ... 
« Previous Showing results 1 — 15 out of 2,115 results