New Lower Bounds for the Maximum Number of Runs in a String
  We show a new lower bound for the maximum number of runs in a string. We prove that for any e > 0, (a -- e)n is an asymptotic lower bound, where a = 56733/60064 = 0.944542. It is superior to the previous bound 0.927 given by Franek et al. Moreover, our construction of the strings and the proof is much simpler than theirs. 
Kazuhiko Kusano, Wataru Matsubara, Akira Ishino, Hideo Bannai, Ayumi   Shinohara