Given the subtractive color dyes currently used in a dye-based color ink jet printer (IJ), it is important to increase the size of the color gamut, especially on high quality coated paper. In a previous study, the author obtained, by means of a computer simulation, the optimum three dyes in order to maximize the size of color gamut. The result showed however, that the color gamut obtained by the optimum three dyes was only 1.1 times larger than that obtained with typical C, M and Y dyes for a dye-based color IJ. This result proves that the typical C, M and Y dyes currently used in a dye-based color IJ achieve good performance for reproducible color gamut. By extension of the previous study, it is now important to know whether an increase from three dyes to four dyes is an efficient way to expand the size of color gamut. In this study we solved for the optimum four dyes for maximizing the size of color gamut, again by means of a computer simulation. The color gamut obtained with the optimum four dyes has been compared with that by the typical C, M and Y dyes for a dye-based color IJ. The result showed that the optimum four dyes achieved a desirable level of performance for reproducible color gamut.