Monochromatic 4-term arithmetic progressions in 2-colorings of Zn

This paper is motivated by a recent result of Wolf on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in Zp, where p is a prime number. Wolf proved that there is a 2-coloring of Zp with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic. In this paper, we present an explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in Zn for general n. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in Zn. Wolf proved that any 2-coloring of Zp has at least (1/16+o(1))p2 monochromatic 4-APs. We improve this lower bound to (7/96+o(1))p2.Our method for Zn naturally apply to the similar problem on [n]. In 2008, Parillo, Robertson, and Saracino constructed a 2-coloring of [n] with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham used a new method to construct a 2-coloring of [n] with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic 5-APs) than random 2-colorings. Our construction gives a 2-coloring of [n] with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs) than random 2-colorings.