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The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is apositive characteristic invariant obtained by comparing the powers of $\a$ withthe Frobenius powers of $J$. We show that under mild assumptions, we can detectthe containment in the integral closure or the tight closure of a parameterideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in termsof the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ arezero-dimensional ideals, and $J$ is generated by a system of parameters. Weprove the conjecture when $J$ is a monomial ideal in a polynomial ring, andalso when $\a$ and $J$ are generated by homogeneous systems of parameters in aCohen-Macaulay graded $k$-algebra.

F-thresholds, tight closure, integral closure, and multiplicity bounds