Let $K=K(V_1,V_2,\cdots,V_p)$ be a $p$-partite complete graph on $n$ vertices and $T_p(n)$ be the Turan graph.
Show that: if $e(K)\geq e(T_r(n))-t$ then $$\sum_{k=1}^p\left(|V_k|-\frac{n}{p}\right)^2\leq 4t.$$
Furthermore, $$|E(K)\Delta E(T_p(n))|\leq 2n\sqrt{\frac{t}{p}}.$$
This statement was claimed in Furedi which can be proved by 'simple calculation'.
I have no idea towards the first inequality. For the second one, I managed to prove a weak one from the first:
$$ 4tp\geq\sum\left(|V_k|-\frac{n}{p}\right)^2\sum 1^2\geq \left(\sum \left||V_k|-\frac{n}{p}\right|\right)^2.$$
Since we need to edit at most $n$ edges for a vertex, then $|E(K) \Delta E(T_p(n))|\leq 2n\sqrt{tp}.$
Any idea is welcomed! :)
Remark: differed by a constant is not important. So we no longer need to juggle with the ceil or floor function in most cases.