  In this paper, we show a direct product theorm in the model of two-party
bounded-round public-coin randomized communication complexity. For a relation f
subset of X times Y times Z (X,Y,Z are finite sets), let R^{(t), pub}_e (f)
denote the two-party t-message public-coin communication complexity of f with
worst case error e. We show that for any relation f and positive integer k:
R^{(t), pub}_{1 - 2^{-Omega(k/t^2)}}(f^k) = Omega(k/t (R^{(t), pub}_{1/3}(f) -
O(t^2))) . In particular, it implies a strong direct product theorem for the
two-party constant-message public-coin randomized communication complexity of
all relations f.
  Our result for example implies a strong direct product theorem for the
pointer chasing problem. This problem has been well studied for understanding
round v/s communication trade-offs in both classical and quantum communication
protocols.
  We show our result using information theoretic arguments. Our arguments and
techniques build on the ones used in [Jain 2011], where a strong direct product
theorem for the two-party one-way public-coin communication complexity of all
relations is shown (that is the special case of our result when t=1). One key
tool used in our work and also in [Jain 2011] is a message compression
technique due to [Braverman and Rao 2011], who used it to show a direct sum
theorem for the two-party bounded-round public-coin randomized communication
complexity of all relations. Another important tool that we use is a correlated
sampling protocol, which for example, has been used in [Holenstein 2007] for
proving a parallel repetition theorem for two-prover games.
