Equations resolving a conjecture of Rado on partition regularity

@article{Alexeev2010EquationsRA,
  title={Equations resolving a conjecture of Rado on partition regularity},
  author={Boris V. Alexeev and Jacob Tsimerman},
  journal={J. Comb. Theory, Ser. A},
  year={2010},
  volume={117},
  pages={1008-1010}
}
Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations
TLDR
It is shown that Fox and Radoicic's family of equations indeed have a degree of regularity of n-1, and a few extensions of this result are proved.
Establishing Conditions on the Degree of Regularity of Linear Homogeneous Equations
In 1933, Rado conjectured that for any positive integer n, there is always a linear homogeneous equation with degree of regularity n. In proving this conjecture, Alexeev and Tsimerman, and
On a Conjecture of Fox and Kleitman on the Degree of Regularity of a Certain Linear Equation
Fox and Kleitman proved in 2006 that for any positive integer b, the 2n-variable equation \(x_1+\cdots +x_n - x_{n+1}- \cdots - x_{2n} \ = \ b\) is not 2n-regular. Moreover, they conjectured the
On a Rado Type Problem for Homogeneous Second Order Linear Recurrences
TLDR
A Ramsey type function S(r;a,b,c) is introduced as the maximum function such that for any $r$-coloring of ${Bbb N} there is a monochromatic sequence satisfying a homogeneous second order linear recurrence.
Regularity of certain diophantine equations
In Ramsey theory, there is a vast literature on regularity questions of linear diophantine equations. Some problems in higher degree have been considered recently. Here, we show that, for every pair
Degree of Regularity of Linear Homogeneous Equations
We define a linear homogeneous equation to be strongly r-regular if, when a finite number of inequalities is added to the equation, the system of the equation and inequalities is still r-regular. In
Erd\H{o}s-Ginzburg-Ziv type generalizations for linear equations and linear inequalities in three variables
For any linear inequality in three variables L, we determine (if it exist) the smallest integer R(L,Z/3Z) such that: for every mapping χ : [1, n] → {0, 1, 2}, with n ≥ R(L,Z/3Z), there is a solution
Computational advances in Rado numbers
TLDR
This thesis presents new methods in the computation of Rado numbers applied to several families of equations and details the application of these tools to various parametrized families of linear and nonlinear equations.
Schur's Theorem and Related Topics in Ramsey Theory
Ramsey theory is a rich field of study and an active area of research. The theory can best be described as a combination of set theory and combinatorics; however, the arguments to prove some of its
A Statement in Combinatorics that is Independent of ZFC (an exposition)
It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which
...
...

References

SHOWING 1-10 OF 15 REFERENCES
On minimal colorings without monochromatic solutions to a linear equation
For a ring R and system L of linear homogeneous equations, we call a coloring of the nonzero elements of R minimal for L if there are no monochromatic solutions to L and the coloring uses as few
Unsolved problems in number theory
TLDR
The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.
Studien zur Kombinatorik
Radoičić, The axiom of choice and the degree of regularity of equations over the reals
  • 2005
E-mail address: balexeev@math.princeton.edu E-mail address: jtsimerm@math.princeton
  • E-mail address: balexeev@math.princeton.edu E-mail address: jtsimerm@math.princeton
Waerden, Beweis einer Baudet’schen Vermutung
  • Nieuw Arch. Wiskunde
  • 1927
Unsolved problems in number theory, third ed., Problem
  • Books in Mathematics, ch. E14,
  • 2004
Uber die Kongruenz x m + y m ≡ z m (mod p), Jahresber Deutsch
  • Math. Verein
  • 1916
The axiom of choice and the degree of regularity of equations over the reals, preprint
  • The axiom of choice and the degree of regularity of equations over the reals, preprint
  • 2005
...
...