# A note on panchromatic colorings

## TL;DR

This paper investigates the minimal size of hypergraphs lacking panchromatic colorings, providing new bounds and explicit constructions, especially for certain ranges of the number of colors and hypergraph uniformity.

## Contribution

It introduces improved lower bounds for large numbers of colors and explicit hypergraph constructions, advancing understanding of panchromatic colorings in hypergraphs.

## Key findings

- New lower bounds for $p(n,r)$ when $r \\geq c \\sqrt{n}$
- Improved upper bounds for $p(n,r)$ when $n = o(r^{3/2})$
- Explicit hypergraph examples with $(\frac{r}{r-1} + o(1))^n$ edges for small $r$

## Abstract

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 $r \leq c \frac{n}{\ln n}$ then all bounds have a type $A_1(n, \ln n, r)(\frac{r}{r-1})^n \leq p(n, r) \leq A_2(n, r, \ln r) (\frac{r}{r-1})^n$, 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 $n = o(r^{3/2})$. Also we show that $p(n,r)$ has upper and lower bounds depend only on $n/r$ when the ratio $n/r$ is small, which can not be reached by the previous probabilistic machinery. Finally we construct an explicit example of a hypergraph without panchromatic coloring and with $(\frac{r}{r-1} + o(1))^n$ edges for $r = o(\sqrt{\frac{n}{\ln n}})$.