  We study $N$-cluster correlation functions in four- and five-dimensional (4D,
5D) bond percolation by extensive Monte Carlo simulation. We reformulate the
transfer Monte Carlo algorithm for percolation [Phys. Rev. E {\bf 72}, 016126
(2005)] using the disjoint-set data structure, and simulate a cylindrical
geometry $L^{d-1}\times \infty$, with the linear size up to $L=512$ for 4D and
$128$ for 5D. We determine with a high precision all possible $N$-cluster
exponents, for $N \! =\!2$ and $3$, and the universal amplitude for a
logarithmic correlation function. From the symmetric correlator with $N \! =
\!2$, we obtain the correlation-length critical exponent as $1/\nu \! =\!
1.4610(12)$ for 4D and $1/\nu \! =\! 1.737 (2)$ for 5D, significantly improving
over the existing results. Estimates for the other exponents and the universal
logarithmic amplitude have not been reported before to our knowledge. Our work
demonstrates the validity of logarithmic conformal field theory and adds to the
growing knowledge for high-dimensional percolation.
