The Kővari-Sós-Turán theorem for $\mathrm{GF}(q)$-representable matroids

In this paper, we establish an analogue of the Kővari-Sós-Turán Theorem for $\mathrm{GF}(q)$-representable matroids. For $2\le s\le t$, we show that if $M$ is a rank-$n$ simple $\mathrm{GF}(q)$-representable matroid having no $M(K_{s,t})$-restriction, then $$\mathrm{\mid }E(M)\mathrm{\mid }=O_{q,s,t}\left(q^{(1-1\mathrm{/}s)n}\right)\mathrm{.}$$ In particular, we prove that the maximum number of elements in a simple rank-$n$ binary matroid with no $M(K_{2,t})$-restriction is ${\mathrm{\Theta }}_t(2^{n\mathrm{/}2})$ where the lower bound is obtained using binary Sidon sets.