We introduce the Bluedisk project, a large program at the Westerbork Synthesis Radio Telescope (WSRT) that has mapped the HI in a sample of 23 nearby galaxies with unusually high HI mass fractions, along with a similar-sized sample of control galaxies matched in stellar mass, size, inclination and redshift. This paper presents the sample selection, observational set-up, data reduction strategy, and a first analysis of the sizes and structural properties of the HI disks. We find that the HI-rich galaxies lie on the same HI mass versus HI size relation as normal spiral galaxies, extending it to total HI masses of $2 times 10^{10} M_{odot}$ and radii R1 of $sim 100$ kpc (where R1 is defined as the radius where the HI column density reaches 1 $M_{odot}$ pc$^{-2}$). HI-rich galaxies have significantly larger values of HI-to-optical size ratio at fixed stellar mass, concentration index, stellar and star formation rate surface density compared to the control sample. The disks of HI-rich galaxies are also significantly more clumpy (i.e. have higher HI Gini and $Delta$Area coefficient) than those of normal spirals. There is no evidence that the disks of HI-rich galaxies are more disturbed: HI-rich galaxies exhibit no difference with respect to control samples in their distributions of HI asymmetry indices or optical/HI disk position angle differences. In fact, the center of the HI distribution corresponds more closely with the center of the optical light in the HI-rich galaxies than in the controls. All these results argue against a scenario in which new gas has been brought in by mergers. It is possible that they may be more consistent with cooling from a surrounding quasi-static halo of warm/hot gas.
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