What systematic relationship did Hubble discover between a galaxy's redshift and its estimated distance?

Recession speed was directly proportional to distance.

After meticulously collecting and analyzing the redshifts ($z$) for numerous galaxies, Edwin Hubble observed a profoundly non-random pattern. He noted that almost every galaxy exhibited a redshift, indicating recession from Earth. Crucially, the magnitude of this redshift, which correlates directly with the speed of recession, was found to be systematically related to how far away the galaxy was located. Specifically, the farther away a galaxy was estimated to be, the greater the observed shift of its spectral lines toward the red end of the spectrum. This linear proportionality between recession velocity ($v$) and distance ($d$) was the empirical evidence suggesting the entire universe was in motion and expanding.