A note on panchromatic colorings

This paper studies the quantity $p(n,r)$, that is the minimal number of edges of an $n$-uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in $r$ colors. If $r\le c\frac{n}{lnn}$ then all bounds have a type $A_1(n,lnn,r){\left(\frac{r}{r?1}\right)}^n\le p(n,r)\le A_2(n,r,lnr){\left(\frac{r}{r?1}\right)}^n$, where $A_1$, $A_2$ are some algebraic fractions. The main result is a new lower bound on $p(n,r)$ when $r$ is at least $c\sqrt{n}$; we improve an upper bound on $p(n,r)$ if $n=o\left(r^{3∕2}\right)$.Also we show that $p(n,r)$ has upper and lower bounds depending only on $n∕r$ when the ratio $n∕r$ is small, which cannot be reached by the previous probabilistic machinery.Finally we construct an explicit example of a hypergraph without panchromatic coloring and with ${(\frac{r}{r?1}+o\left(1\right))}^n$ edges for $r=o\left(\sqrt{\frac{n}{lnn}}\right)$.