When must a set of integers contain an arithmetic progression, and how large can a set without one be? From van der Waerden numbers to density bounds, with small cases open to computer search.
Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.
Let k≥3. Are there k consecutive primes in arithmetic progression?
Prove an asymptotic formula for r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic progression. That is, find f_k with r_k(N) / f_k(N) → 1 as N → ∞.
Estimate h(n) by finding a better upper bound.
What is the largest k such that in any permutation of ℤ there must exist a monotone k-term arithmetic progression x_1 < ⋯ < x_k? Here a permutation of ℤ is a one-sided arrangement a_1, a_2, a_3, … of the integers, i.e.
Must every permutation of ℕ, contain a monotone 4-term arithmetic progression?
Can ℕ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Does the longest arithmetic progression of primes in 1,…,N have length o(log N)?
Let N≥ 1. What is the largest t such that there are A_1,…,A_t⊆ 1,…,N with A_i∩ A_j a non-empty arithmetic progression for all i≠ j?
If A ⊂ ℕ has Σ_n ∈ Afrac 1 n = ∞, then must A contain arbitrarily long arithmetic progressions?
