Mycielski type constructions for hypergraphs associated with fractional colorings

Abstract

It is known that the gap between the clique number and the chromatic number of a graph $G$ can be arbitrarily large. The fractional chromatic number satisfies $\omega (G)\le \chi _f(G) \le \chi (G)$. Larsen et al. (J Graph Theory 19:411–416, 1995) use the Mycielski construction to show that the gap in both of those inequalities can also be arbitrarily large. In this work, we prove the analogous result for the fractional versions of both weak and strong chromatic numbers of uniform hypergraphs. We shall remark that for strong fractional colorings the Mycielski construction and the proof of Larsen et al. can be generalized. For weak fractional colorings, however, the same is not true. We then propose an alternative construction to handle weak fractional colorings of uniform hypergraphs. Finally, we describe, by forbidden induced subgraphs, an interesting class of $r$-uniform hypergraphs with the property that the chromatic number is bounded by a function of its clique number. Such a family arises from affine planes of dimension $r-2$ in $\mathbb {RP}^{d}$.