A family of subsets of [n] is positive linear combination free if the characteristic vector of neither edge is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains (1−o(1))2 subsets of [n] and we give tight bounds on the o(1)2 term. The problem was posed by the late Levon Khachatrian and the resul...

analysis of non-laminar flows in coarse alluvial beds has a wide range of applications in various civil engineering, oil and gas, and geology problems. darcy equation is not valid to analyze transient and turbulent flows, so non-linear equations should be applied. non-linear equations are classified into power and binomial equations. binomial equation is more accurate in a wide range of velocit...

We show that there exist linear $3$-uniform hypergraphs with $n$ vertices and $\Omega(n^2)$ edges which contain no copy of the $3 \times 3$ grid. This makes significant progress on a conjecture Furedi Ruszinko. also discuss connections to proving lower bounds for $(9,6)$ Brown-Erdős-Sos problem Solymosi Solymosi.