On improving a Schur-type theorem in shifted primes

TL;DR

This paper proves that for very large N, any k-coloring of primes less than N guarantees a monochromatic solution to p1 - p2 = p3 - 1, extending Schur-type results to shifted primes.

Contribution

It establishes a new Schur-type theorem for shifted primes with explicit bounds on N for any number of colors.

Findings

Monochromatic solutions exist for large N in k-colorings of shifted primes.

Provides explicit exponential bounds on N depending on k.

Extends classical Schur theorems to shifted prime sets.

Abstract

We show that if $N \geq \exp(\exp(\exp (k^{O(1)})))$, then any $k$-colouring of the primes that are less than $N$ contains a monochromatic solution to $p_1 - p_2 = p_3 -1$.