  Galaxy clusters are the largest and most massive gravitationally bound
structures known in the Universe. Cosmic-Ray (CR) hadrons accelerated at
structure formation shocks and injected by galaxies, are confined in galaxy
clusters where they accumulate for cosmological times. The presence of diffuse
synchrotron radio emission in several clusters proves the existence of
high-energy electrons, and magnetic fields. However, a direct proof of CR
proton acceleration is missing. The presence of CR protons can be probe through
the diffuse gamma-ray emission induced by their hadronic interaction with the
Intra-Cluster Medium (ICM). The Perseus cluster, a nearby cool-core cluster,
has been identified to be among the best candidates to detect such emission. We
present here the results of a very deep observation of the Perseus cluster with
the MAGIC telescopes, accumulating about 250 hours of data from 2009 to 2014.
No evidence of large-scale very-high-energy gamma-ray emission from CR-ICM
interactions has been detected. The derived flux upper limits in the TeV regime
allow us to put stringent constraints on the physics of cluster CRs, in
particular on the CR-to-thermal pressure, the CR acceleration efficiency at
formation shocks and the magnetic field of the central cluster region.
