A note on the &thetas; number of Lovasz and the generalized Delsarte bound

The θ number of Lovasz and the θ_1/2 of Schrijver, McEliece, Rodemich and Rumsey are convex (semidefinite) programming upper bounds on α(G), the size of a maximal independent set of G. It is known that α(G)⩽θ_1/2 (G)⩽θ(G)⩽χ¯(G), where χ¯(G) is the clique cover number of G. The above inequalities suggest that perhaps θ_1/2(G) approximates α(G) from above, and θ(G) approximates χ¯(G) from below for every graph G. Can this approximation be to within a factor of at most n^1-ε for some fixed ε>0? We show, that the following three conjectures are equivalent: 1. ∃ε>0 : θ(G) approximates α(G) for every G within a factor of n^n-ε 2. ∃ε>0 : θ(G) approximates χ¯(G) for every G within a factor of n^n-ε 3. ∃ε>0 : θ_1/2(G) approximates α(G) for every G within a factor of n^n-ε It is not impossible that θ_1/2 approximates χ¯(G), but the latter conjecture looks strictly stronger than 1-3. We give however a simple combinatorial reformulation of this one (we cannot find such for 1-3). We rule out some likely candidates for counterexamples to 1-3 by showing that θ(G) approximates α(G) and χ¯(G) for those graphs G that come from the Hamming scheme