Nearly Erdős-Ko-Rado theorems

Abstract

If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\le n$ then $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ holds by the celebrated Erdős-Ko-Rado theorem. But an intersecting family obviously satisfies the condition $$\left(\genfrac{}{}{0ex}{}{\ell }{2}\right)\le \sum _{1\le i<j\le \ell }|F_i\cap F_j|$$ for any $\ell$ distinct members of the family. It has been proved in [5] that even if $\left(\genfrac{}{}{0ex}{}{\ell }{2}\right)$ is replaced by $\left(\genfrac{}{}{0ex}{}{\ell -1}{2}\right)+1$ the conclusion $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ remains valid for large $n$. However the 1 cannot be omitted, because there is a larger family satisfying that weaker condition. In the present paper we determine the largest size of the family under this weaker condition when $n$ is sufficiently large. All of these are treated in the more general setting of $t$-intersecting families.