On solution-free sets of integers

Abstract

Abstract. Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any 'non-trivial' solutions to L. In this paper we consider the following three general questions:

On solution-free sets of integers II

“…(i) If q divides p and p + q ≤ rq then µ L (n) = ⌈(q − 1)n/q⌉; (ii) If q divides p and p + q ≥ rq then µ L (n) = ⌈(p + q − r)(n − a)/(p + q)⌉ + a where a is the unique non-negative integer 0 ≤ a < q such that n − a is divisible by q; (iii) If q does not divide p, t > 1 and r > (r 1 r 2 − r 1 − r 2 + 4)r 2 r 1 + 1 + r 2 − 1 r 2 1 + (r 1 − 1)(r 2 − 1) then µ L (n) = ⌈(t − 1)n/t⌉. Theorem 2(ii) was already proven (for large enough n) in [14] in the special case when r = 1. (Note though that our work in [14] determines µ L (n) for many equations L not covered by Theorem 2.)…”

“…Theorem 2(ii) was already proven (for large enough n) in [14] in the special case when r = 1. (Note though that our work in [14] determines µ L (n) for many equations L not covered by Theorem 2.) Previously, Hegarty [15] proved Theorem 2(i) in the case when p = q.…”

“…When L is x + y = z it is easy to see that µ L (n) = ⌈n/2⌉ and the interval I n = [⌊n/2⌋ + 1, n] is an extremal set of this size. Recently, the authors [14] established that if L is the equation px + qy = z with p, q ∈ N, p ≥ 2, then µ L (n) = n − ⌊n/(p + q)⌋ for sufficiently large n. Again this bound is attained by the interval I n . Our first result of this paper determines a further class of equations (of the form px + qy = rz) for which I n or T n gives an L-free set of maximum size.…”

“…For other linear equations L, it is also natural to ask whether there are significantly fewer maximal L-free subsets of [n] than there are L-free subsets. In [14] we showed that this is the case for all non-translation-invariant three-variable homogeneous equations L. In particular in this case f max (n, L) ≤ 3 (µ L (n)−µ * L (n))/3+o(n) where µ * L (n) denotes the number of elements in [n] which do not lie in any non-trivial solution to L that only consists of elements from [n]. We also gave other upper bounds on f max (n, L) for equations of the form px + qy = z for fixed p, q ∈ N which in most cases yielded better bounds (see [14]).…”

A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

“…For sum-free subsets, that is A = ( 1 1 −1 ), it is also easy to see that π(A) = 1/2. Hancock and Treglown [16] very recently extended this to matrices of the form A = ( p q −r ) where p, q, r ∈ N such that p ≥ q ≥ r. Unfortunately, unlike the Erdős-Stone-Simonovits Theorem [17,18] in the graph case, no exact characterization of π(A) is known for arbitrary matrices A. However, the following lemma shows that one can still easily bound this value away from 1 for every irredundant and positive matrix.…”