Let $n > k > t geq j geq 1$ be integers. Let $X$ be an $n$-element set, ${Xchoose k}$ the collection of its $k$-subsets. A family $mathcal F subset {Xchoose k}$ is called $t$-intersecting if $|F cap F| geq t$ for all $F, F in mathcal F$. 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 $|mathcal F| = {2k - tchoose k}$. 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., $|mathcal F| > frac32 {2k - tchoose k}$. We provide further improvements for $|mathcal F|$ very large as well.
Download