  Partitioning the edges of a graph into edge disjoint triangles forms a
triangle decomposition of the graph. A famous conjecture by Nash-Williams from
1970 asserts that any sufficiently large, triangle divisible graph on $n$
vertices with minimum degree at least $0.75 n$ admits a triangle decomposition.
In the light of recent results, the fractional version of this problem is of
central importance. A fractional triangle decomposition is an assignment of
non-negative weights to each triangle in a graph such that the sum of the
weights along each edge is precisely 1.
  We show that for any graph on $n$ vertices with minimum degree at least
$0.827327 n$ admits a fractional triangle decomposition. Combined with results
of Barber, K\"{u}hn, Lo, and Osthus, this implies that for every sufficiently
large triangle divisible graph on $n$ vertices with minimum degree at least
$0.82733 n$ admits a triangle decomposition.
