Some progress on open problems

In [1], Dmitry Chistikov, Szabolcs Iván, Anna Lubiw, and Jeffrey Shallit have made an interesting progress on some open problems listed in our book.

A lower bound $\OR(A\times B)\geq \br^*(A)\cdot \OR(B)$, where $\br^*(A)$ is a fractional analogue of the boolean rank of $A$. Problem 7.5 asked to prove (or disprove) a stronger inequality $\OR(A\times B)\geq \br(A)\cdot \OR(B)$, where $\br(A)\geq \br^*(A)$ is the boolean rank of $A$.
A tight estimate $\OR_2(T_n)=n(\lfloor \log_2 n\rfloor+2)-2^{\log_2 n}+1$ on the depth-$2$ OR-complexity of the full triangular matrix $T_n$. This estimate was already proved by Krichevskii [2], but the authors actually prove a stronger fact: all fractional coverings of $T_n$ have large associated cost.
An upper bound $\OR_2(D_n)=O(n^{1.17})$ of the depth-$2$ OR-complexity of the disjointness matrix $D_n$. This is a progress on Problem 7.3. Previously known bounds (as stated in Lemma 4.2 of the book) were $\OR_2(D_n)=\Omega(n^{1.16})$ and $\SUM_2(D_n)=O(n^{1.28})$.
N.B. Note, however, that the latter upper bound is for the SUM-complexity, not for the OR-complexity. It remains unclear whether also this (stronger) upper bound can be improved.

But perhaps more interestingly than numerical bounds themselves is the usage of fractional and greedy coverings.

References:

[1] D. Chistikov, Sz. Iván, A. Lubiw, and J. Shallit, Fractional coverings, greedy coverings, and rectifier networks, arXiv:1509.07588v1.
[2] R.E. Krichevskii, A minimal monotone contact scheme for a Boolean function of n variables, in Diskretnyj Analiz (Discrete Analysis), volume 5, pages 89-92. Institute for Mathematics in the Siberian Section of the Academy of Sciences, Novosibirsk, 1965. In Russian. Paper