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This paper is motivated by a recent result of Wolf \cite{wolf} on the minimumnumber of monochromatic 4-term arithmetic progressions(4-APs, for short) in$\Z_p$, where $p$ is a prime number. Wolf proved that there is a 2-coloring of$\Z_p$ with 0.000386% fewer monochromatic 4-APs than random 2-colorings; theproof is probabilistic and non-constructive. In this paper, we present anexplicit and simple construction of a 2-coloring with 9.3% fewer monochromatic4-APs than random 2-colorings. This problem leads us to consider the minimumnumber of monochromatic 4-APs in $\Z_n$ for general $n$. We obtain both lowerbound and upper bound on the minimum number of monochromatic 4-APs in all2-colorings of $\Z_n$. Wolf proved that any 2-coloring of $\Z_p$ has at least$(1/16+o(1))p^2$ monochromatic 4-APs. We improve this lower bound into$(7/96+o(1))p^2$.  Our results on $\Z_n$ naturally apply to the similar problem on $[n]$ (i.e.,$\{1,2,..., n\}$). In 2008, Parillo, Robertson, and Saracino \cite{prs}constructed a 2-coloring of $[n]$ with 14.6% fewer monochromatic 3-APs thanrandom 2-colorings. In 2010, Butler, Costello, and Graham \cite{BCG} extendedtheir methods and used an extensive computer search to construct a 2-coloringof $[n]$ with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic5-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.

Monochromatic 4-term arithmetic progressions in 2-colorings of $\mathbb Z_n$