Ramsey Type Results on the Solvability of Certain Equation in Zm
نویسنده
چکیده
Csikvári, Gyarmati and Sárközy asked whether there exist Ramsey type theorems for the equations a + b = cd and ab + 1 = cd in Zm for large enough m. In this paper it is proved that for any r-coloring of Zm the more general equation a1 + · · · + an = cd has a nontrivial monochromatic solution. Furthermore, an example is presented which shows that the corresponding statement does not hold for the equation ab + 1 = cd. We reformulate this problem with an additional natural condition, and give a partial positive answer.
منابع مشابه
A note on unique solvability of the absolute value equation
It is proved that applying sufficient regularity conditions to the interval matrix $[A-|B|,A + |B|]$, we can create a new unique solvability condition for the absolute value equation $Ax + B|x|=b$, since regularity of interval matrices implies unique solvability of their corresponding absolute value equation. This condition is formulated in terms of positive deniteness of a certain point matrix...
متن کاملNumerical solvability of system of Fredholm-Hammerstein integral equations using Modification of Hat Function
A system of integral equations can describe different kind of problems in sciences and engineering. There are many different methods for numerical solution of linear and nonlinear system of integral equations. This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integr...
متن کاملOn meromorphic solutions of certain type of difference equations
We mainly discuss the existence of meromorphic (entire) solutions of certain type of non-linear difference equation of the form: $f(z)^m+P(z)f(z+c)^n=Q(z)$, which is a supplement of previous results in [K. Liu, L. Z. Yang and X. L. Liu, Existence of entire solutions of nonlinear difference equations, Czechoslovak Math. J. 61 (2011), no. 2, 565--576, and X. G. Qi...
متن کاملDensity and ramsey type results on algebraic equations with restricted solution sets
In earlier papers Sárközy studied the solvability of the equations a + b = cd, a ∈ A, b ∈ B, c ∈ C, d ∈ D, resp. ab + 1 = cd, a ∈ A, b ∈ B, c ∈ C, d ∈ D where A,B, C,D are “large” subsets of Fp. Later Gyarmati and Sárközy generalized and extended these problems by studying these equations and also other algebraic equations with restricted solution sets over finite fields. Here we will continue ...
متن کاملNILPOTENCY AND SOLUBILITY OF GROUPS RELATIVE TO AN AUTOMORPHISM
In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013