  In this paper we revisit under-determined linear systems of equations with
sparse solutions. As is well known, these systems are among core mathematical
problems of a very popular compressed sensing field. The popularity of the
field as well as a substantial academic interest in linear systems with sparse
solutions are in a significant part due to seminal results
\cite{CRT,DonohoPol}. Namely, working in a statistical scenario,
\cite{CRT,DonohoPol} provided substantial mathematical progress in
characterizing relation between the dimensions of the systems and the sparsity
of unknown vectors recoverable through a particular polynomial technique called
$\ell_1$-minimization. In our own series of work
\cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10} we also provided a
collection of mathematical results related to these problems. While, Donoho's
work \cite{DonohoPol,DonohoUnsigned} established (and our own work
\cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10} reaffirmed) the typical or
the so-called \emph{weak threshold} behavior of $\ell_1$-minimization many
important questions remain unanswered. Among the most important ones are those
that relate to non-typical or the so-called \emph{strong threshold} behavior.
These questions are usually combinatorial in nature and known techniques come
up short of providing the exact answers. In this paper we provide a powerful
mechanism that that can be used to attack the "tough" scenario, i.e. the
\emph{strong threshold} (and its a similar form called \emph{sectional
threshold}) of $\ell_1$-minimization.
