For prime power t and n = t^{2e-1}, ex(n, K_{t,t}, K_{2,t+1}) = (1 + o(1)) n² / (2t(t-1)).
$K_{2,t+1}$-free graphs with many copies of $K_{t,t}$
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abstract
For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=\Theta_t(n^2). \] This answers a question of Spiro.
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2026 1verdicts
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$K_{2, t+1}$-free graphs containing an optimal number of $K_{t, t}$'s
For prime power t and n = t^{2e-1}, ex(n, K_{t,t}, K_{2,t+1}) = (1 + o(1)) n² / (2t(t-1)).