The predicted piecewise-quadratic formula for the Ramsey-Tur\e1n density

Let $f\colon(0,1]\longrightarrow\mathds R$ be the Ramsey\e1Tur\e1n density function. For $k\ge2$, define

$$f_k(\alpha)=k(k-1)-2k(3k-4)\alpha+(3k-4)(3k-1)\alpha^2.$$

Ramsey\e1Tur\e1n density formula. The function $f$ is given by

$$f(\alpha)=\begin{cases} \alpha,&\alpha\le\frac13,\\ f_k(\alpha),&\frac{k}{3k-1}\le\alpha<\frac{k-1}{3k-4}\text{ for some }k\ge2,\\ \frac12,&\frac12\le\alpha\le1. \end{cases}$$

This is the precise prediction for the density function that would follow from the preceding extremal conjecture. Thus its status is tied to the still-open range $\alpha\in(\frac13,\frac38)$, while the portions corresponding to the established cases are known.