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A density statement generalizing Schur's theorem

1986
*
Journal of combinatorial theory. Series A
*

The straightforward

doi:10.1016/0097-3165(86)90074-9
fatcat:3nayp2sfgvb3xm6daygbhsw7pq
*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. ...##
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Density versions of two generalizations of Schur's theorem

1988
*
Journal of combinatorial theory. Series A
*

Given

doi:10.1016/0097-3165(88)90072-6
fatcat:bgs6bmze3raj5m7vrjbb27qtx4
*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*. ...##
###
Fermat's Last Theorem Implies Euclid's Infinitude of Primes
[article]

2020
*
arXiv
*
pre-print

We show that Fermat's last

arXiv:2009.06722v2
fatcat:juojyc4enfgtvhagiuetxb2ypm
*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. ...##
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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*...##
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Splitting of integer polynomials over fields of prime order
[article]

2018
*
arXiv
*
pre-print

It is well known that

arXiv:1802.10562v2
fatcat:dxqbzw4pnzdopjo2q2b4dh7r4m
*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 . ...##
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The Ramsey-type version of a problem of Pomerance and Schinzel

2012
*
Acta Arithmetica
*

The author is greateful to

doi:10.4064/aa156-1-1
fatcat:5rtadv3n4fhvngpva3sifkqlgq
*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. ...##
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Curves of Finite Total Curvature
[article]

2007
*
arXiv
*
pre-print

To explore these ideas, we consider

arXiv:math/0606007v2
fatcat:cnh7g6rtyvg5td2zymemeekeji
*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. ...##
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Remarks on the quantum de Finetti theorem for bosonic systems
[article]

2014
*
arXiv
*
pre-print

The quantum de Finetti

arXiv:1310.2200v3
fatcat:pkmaxhqpmbcuzeugfozkcdpif4
*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 . ...##
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SCHUR'S COLOURING THEOREM FOR NONCOMMUTING PAIRS

2019
*
Bulletin of the Australian Mathematical Society
*

to contain

doi:10.1017/s0004972719000406
fatcat:43p66nz44fcwvhj66tunlzwam4
*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. ...##
###
Solvability of Rado systems in D-sets
[article]

2008
*
arXiv
*
pre-print

(Since one cell of any finite partition is central, this

arXiv:0809.2281v1
fatcat:yt7waoozdrevpa6sa2zmjzcwcy
*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*. ...##
###
Remarks on the Quantum de Finetti Theorem for Bosonic Systems

2014
*
Applied Mathematics Research eXpress
*

The quantum de Finetti

doi:10.1093/amrx/abu006
fatcat:tuedebblerey7cpbncdslc3ljy
*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 . ...##
###
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

2009
*
Topology and its Applications
*

(Since one cell of any finite partition is central, this

doi:10.1016/j.topol.2009.04.019
fatcat:3l6fsc4thjbbrgbczlpgfsbsty
*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*. ...##
###
Extremal results for random discrete structures

2016
*
Annals of Mathematics
*

In particular, we verify

doi:10.4007/annals.2016.184.2.1
fatcat:h37q3qgncrbepotos7k36yo3ha
*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 ...##
###
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. ...

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