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Source. Published p. 261, following Corollary 1.3 (PDF).

Statement. Fix $0<\rho<1$ and let $l=\lfloor\rho n\rfloor$. There is a family avoiding intersection $l$, even for all cross pairs including equal members, with

$$|\mathcal F_n| =\exp\left(nh\left(\frac{1+\rho}{2}\right)+O_\rho(\log(n+1))\right).$$

In particular its exponential base, as $\rho\to0$ after $n\to\infty$, is $2-\rho^2+O(\rho^4)$.

Proof. Put $k=\lfloor(n+l)/2\rfloor+1$ and take all sets of size at least $k$. Every cross intersection has size at least $2k-n>l$. Since $k>n/2$, the family size lies between $\binom nk$ and $(n+1)\binom nk$. The factorial estimate and $k/n=(1+\rho)/2+O(1/n)$ give the displayed formula. Finally $h$ is smooth and symmetric around $1/2$, and Taylor expansion gives $h((1+\rho)/2)=\log2-\rho^2/2+O(\rho^4)$. Exponentiating yields the stated base. $\square$

Dependencies. entropy_estimates.