There may not be any specific federal law prohibiting pyramid schemes in the US, but there are some state laws prohibiting them, and the FTC prosecutes people engaging in pyramid schemes under other laws.

At the Commission, we bring cases against pyramid schemes under the FTC Act, which broadly prohibits "unfair or deceptive acts or practices in or affecting commerce."(12) That Act allows the Commission to file suit in federal court and seek a variety of equitable remedies, including injunctive relief, a freeze over the defendants' assets, a receivership over the defendants' business, and redress or restitution for consumers.

FTC Website