What's the problem? Decades ago, Arp started to observe various galaxies and quasars. See, for example
In various individual cases, it looks like there is a galaxy connected to a quasar by a "gas filament". It really looks so, especially if you glance at the representative pictures available on the link above. The optically smaller components of this "connected multiobject" always have higher redshifts than the "large" component.
Well, there is a "subtle" discrepancy with our conventional wisdom. The quasars should be much further than the galaxies. Well, at least this is what we deduce, in agreement with the Big Bang theory, from the redshifts: the quasars (which are the smaller objects on the pictures) always have a greater redshifts than the galaxies.
Obviously, if the redshift is a Doppler redshift induced by the velocity from the expanding Universe, i.e. if it is cosmological in origin, it's clear that the quasars - conventionally believed to be as powerful as dozens of galaxies - should be much further than the galaxies around. If they're so much further, hundreds of millions of light years, they should not be connected with a "filament", right?
In the conventional Big Bang picture, all these filaments must be just some coincidences, pieces of gas that happen to fit, optically, between the galaxy and the quasar, but that are highly separated in the radial dimension. The main quantitative question about Arp's evidence is, of course, whether it is statistically plausible that he observes his (pretty large) number of these filaments. Note that it is just "slightly" controversial to say that the quasars seem to be "unusually" close to a galaxy in most cases.
OK, so these filaments should all be coincidences. Except, of course, the possibility that Arp is correct and these connections are real. In that case, the incorporation of the quasars to the Big Bang model (i.e. the relation between their redshift and their distance) would have to be sacrificed, and perhaps a more radical change would have to follow.
In a conservative version of this hypothetical revolution, the quasars could be some rather heavy objects with a huge gravitational redshift. Correct me if I'm wrong, but this object could be nothing that we normally "know" in the Universe. In Arp's description, the quasars are "ejected" from the galaxies, and therefore the redshift is a Doppler redshift, but not a cosmological one. Well, I leave the other "progressive" versions of this revolution to others.
There are rumors that some conclusions of Arp, who had to move to Germany, result from clear mathematical errors. I have not been able to check any of these statements.
If someone wants to see some very recent papers advocating Arp's point of view by a "consistency check", see e.g. astro-ph/0409025.