Lower-tail and fixed nonzero claw-density variational problems

Solve the lower-tail variational problem for claws and the variational problem for a fixed nonzero claw density.

Background

The paper solves variational problems for claw-freeness, corresponding to zero induced claw density, including fixed edge density and the large-deviation rate for the event that G(n,p) is claw-free.

It explicitly identifies two extensions that remain unresolved: the lower-tail problem for the number of claws and the constrained problem in which the induced claw density is fixed at a positive, nonzero value. The authors state that these problems are likely to require new techniques.