More about the uniformity of the CMBR:
The cosmic microwave background is the afterglow radiation left over from the hot Big Bang. Its temperature is extremely uniform all over the sky.
However, standard Big Bang theory does not account for all of the observed properties of the CMB. In particular, once we remove the dipole that arises due to our motion in the Universe, the CMB is incredibly uniform across the sky, varying by no more than one part in ten thousand. This suggests that regions of the Universe that are now widely separated, were once close enough to ‘communicate’ with each other in order to equalise their temperature. However, this is not possible given standard Big Bang theory, the age of the Universe, and the finite speed of light.
As for the universe being Euclidean, this is the flatness problem.
In the case of the flatness problem, the parameter which appears fine-tuned is the density of matter and energy in the universe. This value affects the curvature of space-time, with a very specific critical value being required for a flat universe. The current density of the universe is observed to be very close to this critical value. Since any departure of the total density from the critical value would increase rapidly over cosmic time, the early universe must have had a density even closer to the critical density, departing from it by one part in 10^62 or less. This leads cosmologists to question how the initial density came to be so closely fine-tuned to this ‘special’ value.
Right now, we’ve only measured the curvature to a level of 1-part-in-400, and find that it’s indistinguishable from flat.
so far most cosmological measurements seem to favor a flat universe.