Work

Monochromatic solutions to x+y=z2 in the interval [N,cN4]

Year: 2018

Type: article

Abstract: Green and Lindqvist proved that for any 2-colouring of N, there are infinitely many monochromatic solutions to x + y = z 2 . In fact, they showed the existence of a monochromatic solution in every interval [ N , c N 8 ] with large enough N. In this short note we give a different proof for their theorem and prove that a monochromatic solution exists in every interval [ N , 10 4 N 4 ] with large enough N. A 2-colouring of [ N , ( 1 / 27 ) N 4 ] avoiding monochromatic solutions to x + y = z 2 is also presented, which shows that in 10 4 N 4 only the constant factor can be reduced.

Source: Bulletin of the London Mathematical Society

DOI: 10.1112/blms.12207

Author: Péter Pál Pach

Institutions: Budapest University of Technology and Economics, University of Warwick

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: European Research Council

Award: G3210981189