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

This paper is motivated by a recent result of Wolf [12] 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 and non-constructive. 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 4APs in Zn for general n. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in all 2-colorings of Zn. Wolf proved that any 2-coloring of Zp has at least (1/16 + o(1))p 2 monochromatic 4-APs. We improve this lower bound into (7/96+o(1))p. Our results on Zn naturally apply to the similar problem on [n] (i.e., {1, 2, . . . , n}). In 2008, Parillo, Robertson, and Saracino [5] constructed a 2-coloring of [n] with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham [1] extended their methods and used an extensive computer search to construct a 2coloring 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.