$F$-thresholds of filtrations of ideals

In this article, we extend the notion of the $F$-thresholds of ideals to the $F$-thresholds for filtrations of ideals. The existence of $F$-thresholds of filtrations are established for various types of filtrations. Moreover, various necessary and sufficient conditions for finiteness of $F$-thresholds are given. We also give some effective upper bounds of the symbolic $F$-thresholds. A sufficient condition for symbolic $F$-splitting in terms of its symbolic $F$-threshold is given. We compute the symbolic $F$-thresholds of various determinantal ideals, $F$-K\"onig ideals, and many more.