Question 6
Let $\mathcal H$ be a family of nonempty subsets of $\{1,2,3,\ldots,2027\}$ with the following property: for every $A\in\mathcal H$ and every subset $B\subset A$, one has $B\in\mathcal H$. Let $\ell_{\mathcal H},c_{\mathcal H}$ denote, respectively, the numbers of sets in $\mathcal H$ having an even number of elements and an odd number of elements. Prove that
$$\ell_{\mathcal H}-c_{\mathcal H}\le\binom{2026}{1013}.$$
$$\ell_{\mathcal H}-c_{\mathcal H}\le\binom{2026}{1013}.$$