  The celebrated Hadwiger's conjecture states that if a graph contains no
$K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular
imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set
of size at least $n/t$. In 1982, Duchet and Meyniel proved that this bound
holds within a factor $2$. Their bound has been improved; most notably in an
absolute factor by Fox, which was later improved by Balogh and Kostochka. Here
we consider the same question for triangle-free graphs. By the results of
Shearer and Kostochka and Thomason, it follows that any triangle-free graph
with no $K_t$ minor has an independent set of size
$\Omega(\tfrac{\sqrt{\log{t}}}{t}n)$. We show that a much larger independent
set exists; for all sufficiently large $t$ every triangle-free graph on $n$
vertices with no $K_t$-minor has an independent set of size $
\tfrac{n}{t^{1-\varepsilon}}$. This answers a question of Sergey Norin.
