Borsuk's Conjecture Fails in Dimensions 321 and 322
  Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323.   Here we show that Borsuk's conjecture fails in dimensions 321 and 322. (This result has been independently discovered by Hinrichs and Richter.) 
Oleg Pikhurko