Document Type : Research

Author

• amir khamseh

Department of Mathematics, Kharazmi University

Abstract

Hindman’s Theorem states that for every coloring of natural numbers N with finitely many colors, there is an infinite set H such that the set of numbers which can be written as a sum of distinct elements of H is monochromatic. On the other hand, Brauer’s Theorem states that for all r,l,s≥1, there exists t=t(r,l,s) such that if the interval [1,t] is r-colored then there exists a,b>0 such that the set \{a,b,a+b,a+2b,…,a+(l-1)b\}⊆[1,t] is monochromatic. If A and B are sets, FS^A (B) is the set of all sums of j-many distinct elements of B, for all j∈A. Hindman-Brauer Theorem is the following statement: for every r-coloring of the set of natural numbers N, there is an infinite set H⊆N and a,b>0 such that FS^(\{a,b,a+b,a+2b,…,a+(l-1)b\}) is monochromatic. In this paper, we study the finite version of Hindman-Brauer Theorem and also Hindman-Schur Theorem and show that these results are provable in first order Peano Arithmetic. Also, we will see that these results are provable if we consider the apartness condition.

Keywords

• Provability
• Peano Arithmetic
• Hindman Theorem, Brauer Theorem, Schur Theorem,
• Ramsey Theorem

Main Subjects

• Standard Mathematical Logic