On strengthenings of the intersecting shadow theorem

Abstract

Let $n>k>t\ge j\ge 1$ be integers. Let $X$ be an $n$-element set, $\left(\genfrac{}{}{0ex}{}{X}{k}\right)$ the collection of its $k$-subsets. A family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{X}{k}\right)$ is called $t$-intersecting if $|F\cap F^{\prime }|\ge t$ for all $F,F^{\prime }\in \mathcal{F}$. The $j$'th shadow ${\partial }^j\mathcal{F}$ is the collection of all $(k-j)$-subsets that are contained in some member of~$\mathcal{F}$. Estimating $|{\partial }^j\mathcal{F}|$ as a function of $|\mathcal{F}|$ is a widely used tool in extremal set theory. A classical result of the second author (Theorem \ref{th:1.3}) provides such a bound for $t$-intersecting families. It is best possible for $|\mathcal{F}|=\left(\genfrac{}{}{0ex}{}{2k-t}{k}\right)$. Our main result is Theorem \ref{th:1.4} which gives an asymptotically optimal bound on $|{\partial }^j\mathcal{F}|/|\mathcal{F}|$ for $|\mathcal{F}|$ slightly larger, e.g., $|\mathcal{F}|>\frac{3}{2}\left(\genfrac{}{}{0ex}{}{2k-t}{k}\right)$. We provide further improvements for $|\mathcal{F}|$ very large as well.