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{\bf B\'{e}la Bollob\'{a}s and Vladimir Nikiforov}
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{\bf The Sum of Degrees in Cliques}
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For every graph $G,$ let
$$
\Delta_{r}\left(G\right)  =\max\left\{  \sum_{u\in R}d\left(  u\right)
:R\hbox{ is an }r\hbox{-clique of }G\right\}
$$
and let $\Delta_{r}\left(  n,m\right)  $ be the minimum of $\Delta_{r}\left(
G\right)$ taken over all graphs of order $n$ and size $m$. Write
$t_{r}\left(  n\right)  $ for the size of the $r$-chromatic Tur\'{a}n graph of
order $n$.

Improving earlier results of Edwards and Faudree, we show that for every
$r\geq2,$ if $m\geq t_{r}\left(  n\right)$, then
$$
\Delta_{r}\left(  n,m\right)  \geq\frac{2rm}{n},\eqno{(1)}
$$
as conjectured by Bollob\'{a}s and Erd\H{o}s.

It is known that inequality (1) fails for $m<t_{r}\left(n\right)$. 
However, we show that for every $\varepsilon>0,$ there is $\delta>0$ such
that if $m>t_{r}\left(  n\right)  -\delta n^{2}$ then
$$
\Delta_{r}\left(  n,m\right)  \geq\left(  1-\varepsilon\right)  \frac{2rm}{n}.
$$

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