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Let delta k be such that in any 2 -colouring of 1, ldots,n there exist at least ( delta k+o(1))n^2 many monochromatic k -term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) f…
Let N geq 1 . What is the largest t such that there are A 1, ldots,A t subseteq 1, ldots,N with A i cap A j a non-empty arithmetic progression for all i neq j ?
Let A(n)= a 0<a 1< cdots be the sequence defined by a 0=0 and a 1=n , and for k geq 1 define a k+1 as the least positive integer such that there is no three-term arithmetic progression in $ a 0, ldots,a k…
Let G k(N) be such that any set of N integers contains a subset of size at least G k(N) which does not contain a k -term arithmetic progression. Determine the size of G k(N) . How does it relate to R k(N) , the …
Does the longest arithmetic progression of primes in 1, ldots,N have length o( log N) ?
Can mathbb N be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Must every permutation of mathbb N contain a monotone 4-term arithmetic progression? In other words, given a permutation x of mathbb N must there be indices with either i<j<k<l or i j k l such that $x i,x j…
What is the largest k such that in any permutation of mathbb Z there must exist a monotone k -term arithmetic progression x 1< cdots<x k ?
Find the best function f(d) such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference d of length f(d) for infinitely many d .
Find the smallest h(d) such that the following holds. There exists a function f: mathbb N to -1,1 such that, for every d geq 1 , max P d left lvert sum n in P d f(n) right rvert leq h(d), where $P d…
Let N(k, ell) be the minimal N such that for any f: 1, ldots,N to -1,1 there must exist a k -term arithmetic progression P such that left lvert sum n in P f(n) right rvert geq ell. Find good up…
Let k geq 3 and f(k) be the supremum of sum n in A frac 1 n as A ranges over all sets of positive integers which do not contain a k -term arithmetic progression. Estimate f(k) . Is lim k to infty …
Let h(N) be the smallest k such that 1, ldots,N can be coloured with k colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate h(N) .
Let r k(N) be the largest possible size of a subset of 1, ldots,N that does not contain any non-trivial k -term arithmetic progression. Prove an asymptotic formula for r k(N) .
Let k geq 3 . Are there k consecutive primes in arithmetic progression?