Bootstrapping partition regularity of linear systems. Issue 3 (9th August 2020) - Sanders, Tom
Abstract: Suppose that A is a k × d matrix of integers and write _A:N→ N{ ∞ } for the function taking r to the largest N such that there is an r -colouring C of [ N ] with _C CC^dA = . We show that if ℜ A ( r ) < ∞ for all rN then R_A(r) ((r^O_A(1))) for all r ⩾ 2. When the kernel of A consists only of Brauer configurations – that is, vectors of the form ( y, x, x + y, …, x + ( d − 2) y ) – the above statement has been proved by Chapman and Prendiville with good bounds on the O A (1) term.