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We show that for any 
$\varepsilon \gt 0$ and 
$\Delta \in \mathbb{N}$, there exists 
$\alpha \gt 0$ such that for sufficiently large 
$n$, every 
$n$-vertex graph 
$G$ satisfying that 
$\delta (G)\geq \varepsilon n$ and 
$e(X, Y)\gt 0$ for every pair of disjoint vertex sets 
$X, Y\subseteq V(G)$ of size 
$\alpha n$ contains all spanning trees with maximum degree at most 
$\Delta$. This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.
$\boldsymbol\ell$
-independence number
Given a graph
$G$
and an integer
$\ell \ge 2$
, we denote by
$\alpha _{\ell }(G)$
the maximum size of a
$K_{\ell }$
-free subset of vertices in
$V(G)$
. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in
$n$
-vertex graphs
$G$
with
$\alpha _{\ell }(G) = o(n)$
, which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for
$K_r$
-factors in
$n$
-vertex graphs
$G$
with
$\alpha _\ell (G)=n^{1-o(1)}$
for all
$r\ge \ell \ge 2$
.
