A Simple proof of the Gan-Loh-Sudakov Conjecture

Abstract

We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $\Delta$ has no more than $a\binom{\Delta+1}{t}+\binom{b}{t}$ cliques of size $t \ (t \ge 3)$, where $n = a(\Delta+1)+b \ (0 \le b \le \Delta)$.

Generalized Turán problems for double stars

“…See [12] for more on generalized Turán problems with constant values. Another example is ex(s, K r , S k ), where the extremal graph consists of ⌊s/k⌋ copies of K k and a clique on the remaining vertices [4]. If s − k⌊s/k⌋ < r, then those additional vertices are useless.…”