On colouring integers avoiding \(t\)-AP distance-sets

Abstract

A \(t\)-AP is a sequence of the form \(a,a+d,\ldots, a+(t-1)d\),
where \(a,d\in \mathbb{Z}\). Given a finite set \(X\) and positive integers \(d\), \(t\), \(a_1,a_2,\ldots,a_{t-1}\), define \(\nu(X,d) = |\{(x,y):{x,y\in{X}},{y>x},{y-x=d}\}|\), \((a_1,a_2,\ldots,a_{t-1};d) =\) a collection \(X\) s.t. \(\nu(X,d\cdot{i})\geq a_i\) for \(1\leq i\leq t-1\).

In this paper, we investigate
the structure of sets with bounded number of pairs with certain gaps.
Let \((t-1,t-2,\ldots,1; d)\) be called a \emph{\(t\)-AP distance-set} of size at least \(t\).
A \(k\)-colouring of integers \(1,2,\ldots, n\) is a mapping \(\{1,2,\ldots,n\}\rightarrow \{0,1,\ldots,k-1\}\) where
\(0,1,\ldots,k-1\) are colours.
Let \(ww(k,t)\) denote the
smallest positive integer \(n\) such that every \(k\)-colouring of \(1,2,\ldots,n\)
contains a monochromatic \(t\)-AP distance-set for some \(d>0\).
We conjecture that \(ww(2,t)\geq t^2\) and prove the lower bound for most cases.
We also generalize the notion of \(ww(k,t)\) and prove several lower bounds.