Dependence of the degree bound on gamma and t

Determine how the smallest function d(gamma,t) depends on the max-norm bound gamma and the forbidden-submatrix parameter t for Boolean matrices with gamma_2-norm at most gamma and no t by t all-ones submatrix.

Background

The paper proves that for every gamma greater than 1 and integer t at least 2, Boolean matrices with gamma_2-norm at most gamma and no t by t all-ones submatrix have average degree at most d(gamma,t). For t equal to 2, the paper obtains the sharp estimate d(gamma,2)=O(gamma^2).

For general t, the proof yields only d(gamma,t)=t to the O(gamma^8), while the authors state that they believe the dependence on t should be linear. A construction in the same section gives an exponential lower bound in gamma, showing that the quantitative problem is nontrivial in both parameters.