Excluding a line from gammoids

Source abstract

For all positive integers $\ell$ and $r$, we prove that a finite simple rank-$r$ gammoid with no $U_{2,\ell+2}$-minor has at most $\ell(r-1)+1$ elements. This gives an affirmative answer to Problem~7.2 of Boretsky and Walsh [European J. Combin.\ 137 (2026), 104419], for both gammoids and transversal matroids. The bound is sharp for every $\ell$ and $r$. The key estimate is that, for every finite simple gammoid $M$, the sum of $|L|-2$ over its long lines is at most $|E(M)|-r(M)$.