A separation between tropical matrix ranks

We continue to study the rank functions of tropical matrices. In this paper, we explain how reduce computation ranks for matrices over ‘supertropical semifield’ standard case. Using a counting approach, prove existence 01 01 -matrix with many ones and without large all-one submatrices, put our results together construct an alttext="n times n"> n × encoding="application/x-tex">n\times n matrix alttext="o left-parenthesis n Superscript 0.5 plus epsilon Baseline right-parenthesis"> o ( 0.5 + ε stretchy="false">) encoding="application/x-tex">o(n^{0.5+\varepsilon }) Kapranov minus o −<!-- − encoding="application/x-tex">n-o(n) .