Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ colors bounded above by $\beta n$ with some fixed $\beta>1$?

A positive answer to this question will yield a positive answer to http://mathoverflow.net/questions/212268/a-constrained-minimum-edge-coloring.