This paper shows that when you color the integers with three colors, the threshold at which a long monochromatic arithmetic progression must appear grows faster than any exponential function of the progression length. More concretely, for all sufficiently large k the three-color van der Waerden number w(k;3) exceeds 2^{k (log^* k)/4}. Here w(k;3) is the smallest N such that every coloring of {1,…,N} with three colors contains a monochromatic k-term arithmetic progression (a sequence of k equally spaced numbers all the same color), and log^* k is the iterated logarithm — the number of times you must apply the logarithm to k before the result is at most 1. Because log^* grows extremely slowly but unboundedly, the result implies super-exponential growth in k for three colors.

To get this bound the authors build explicit colorings that avoid long monochromatic arithmetic progressions. A key ingredient is a probabilistic construction of dense subsets of large cyclic groups that contain no k-term arithmetic progression. They then use a new “random shifted product” construction to combine smaller examples into larger ones. Each step multiplies the size of the construction by an amount exponential in k, at the cost of a large loss in the relative density of the special color. Repeating this roughly (log^* k)/2 times yields the stated lower bound on w(k;3).

The paper also proves stronger lower bounds in other regimes. When the number of colors r is large compared with log k, the authors show the multicolor van der Waerden number satisfies a bound of the form w(k;r) ≥ r^{(1−ε)k log k} for r at least about (log k)^{3/ε} and k large enough. They further obtain a bound for the canonical van der Waerden number H(k) — the smallest N so that every coloring of {1,…,N} has either a monochromatic k-term progression or a rainbow (all different colors) k-term progression — giving H(k) ≥ k^{(1−o(1))k log k}. The latter resolves a problem of Erdős and Graham mentioned in the paper.