Every family of at most \(2^{d-1}\) \(d\)-sets is \(2\)-colorable, that is, \(m(d) {\gt} 2^{d-1}\).

Every family of at most \(2^{d-1}\) \(d\)-sets is \(2\)-colorable, that is, \(m(d) {\gt} 2^{d-1}\).

For every \(k \geq 2\), there exists a graph \(G\) with chromatic number \(\chi (G) {\gt} k\) and girth \(\gamma (G) {\gt} k\).

Let \(G\) be a simple graph with \(n\) vertices and \(m\) edges, where \(m \ge 4n\). Then