= de Caen bound for hypergraph Turán density
{c}
{title2=$\pi(K_r^\ell)\le1-\binom{r-1}{\ell-1}^{-1}$}
For the complete $\ell$-uniform hypergraph $K_r^\ell$, with $r\ge\ell\ge2$, the bound is $\pi(K_r^\ell)\le1-1/\binom{r-1}{\ell-1}$. In the complementary covering formulation, every $r$-set containing an edge forces asymptotic edge density at least $1/\binom{r-1}{\ell-1}$. It follows from clique-extension incidence inequalities and <Cauchy-Schwarz inequality> estimates on link degrees.
Back to article page
