  In this paper we introduce a general framework for proving lower bounds for
various Ramsey type problems within random settings. The main idea is to view
the problem from an algorithmic perspective: we aim at providing an algorithm
that finds the desired colouring with high probability. Our framework allows to
reduce the probabilistic problem of whether the Ramsey property at hand holds
for random (hyper)graphs with edge probability $p$ to a deterministic question
of whether there exists a finite graph that forms an obstruction.
  In the second part of the paper we apply this framework to address and solve
various open problems. In particular, we extend the result of Bohman, Frieze,
Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs to
the case of $2$ colors and to hypergraph cliques. As a corollary, this proves a
matching lower bound for the result of Friedgut, R\"odl and Schacht (2010) and,
independently, Conlon and Gowers (2014+) for the classical Ramsey problem for
hypergraphs in the case of cliques. Finally, we provide matching lower bounds
for a proper-colouring version of anti-Ramsey problems introduced by
Kohayakawa, Konstadinidis and Mota~(2014) in the case of cliques and cycles.
