  Galaxy clusters are being assembled today in the most energetic phase of
hierarchical structure formation which manifests itself in powerful shocks that
contribute to a substantial energy density of cosmic rays (CRs). Hence,
clusters are expected to be luminous gamma-ray emitters since they also act as
energy reservoirs for additional CR sources, such as active galactic nuclei and
supernova-driven galactic winds. To detect the gamma-ray emission from CR
interactions with the ambient cluster gas, we conducted the deepest to date
observational campaign targeting a galaxy cluster at very high-energy
gamma-rays and observed the Perseus cluster with the MAGIC Cherenkov telescopes
for a total of ~85 hr of effective observing time. This campaign resulted in
the detection of the central radio galaxy NGC 1275 at energies E > 100 GeV with
a very steep energy spectrum. Here, we restrict our analysis to energies E >
630 GeV and detect no significant gamma-ray excess. This constrains the average
CR-to-thermal pressure ratio to be <= 1-2%, depending on assumptions and the
model for CR emission. Comparing these gamma-ray upper limits to predictions
from cosmological cluster simulations that include CRs constrains the maximum
CR acceleration efficiency at structure formation shocks to be < 50%.
Alternatively, this may argue for non-negligible CR transport processes such as
CR streaming and diffusion into the outer cluster regions. Finally, we derive
lower limits on the magnetic field distribution assuming that the Perseus radio
mini-halo is generated by secondary electrons/positrons that are created in
hadronic CR interactions: assuming a spectrum of E^-2.2 around TeV energies as
implied by cluster simulations, we limit the central magnetic field to be > 4-9
microG, depending on the rate of decline of the magnetic field strength toward
larger radii.
