A symmetric basis exchange replaces two matroid bases 
and 
by
 |
where 
,
,
and both resulting sets are matroid bases. Each
element therefore occurs the same number of times in the pair before and after the
exchange.
The symmetric basis exchange theorem states that, for every pair of matroid bases and every 
, at least one suitable 
exists. This local existence statement does not imply that
repeated exchanges connect all pairs with the same multiset
union, as proposed in White's conjecture.
