In this paper we consider relations between chaos in the sense of Li and Yorke, and $\omega$-chaos. The main aim is to show how important is the size of scrambled sets in definitions of chaos. We provide an example of an $\omega$-chaotic map on a compact metric space which is chaotic in the sense of Li and Yorke, but any scrambled set contains only two points. Chaos in the sense of Li and Yorke cannot be excluded: We show that any continuous map of a compact metric space which is $\omega$-chaotic, must be chaotic in the sense of Li and Yorke. Since it is known that, for continuous maps of the interval, Li and Yorke chaos does not imply $\omega$-chaos, Li and Yorke chaos on compact metric spaces appears to be weaker. We also consider, among others, the relations of the two notions of chaos on countably infinite compact spaces.
The paper will be presented at the conference SVOC 2002. The author's recent work "Scrambled sets for transitive maps", which was presented at SVOC 2001, and which will appear in Real Analysis Exchange, also involves $\omega$-chaos, but the results neither are related to, nor are used in the present paper.
Keywords. $\omega$-chaos, Li and Yorke chaos, scrambled sets.