Refinements of Alon-Babai-Suzuki-type intersection theorems via non-shadows and binomial support

Abstract: We prove a multilevel non-shadow refinement of the Alon--Babai--Suzuki (ABS) nonuniform restricted-intersection theorem. Let $K={k_1,\dots,k_r}$ and let $L$ be a set with $|L|=s$. If $$\mathcal{F}\subseteq \bigcup_{k\in K}\binom{[n]}{k}$$ is $L$-intersecting and $k_i&gt;s-r$ for every $i$, then $$|\mathcal{F}| + \sum_{j=s-r+1}^{s} |\mathcal{N}<em>j(\mathcal{F})| \le N(n,s,r),$$ equivalently $$|\mathcal{F}| \le \sum</em>{j=s-r+1}^{s} |\partial_j\mathcal{F}|.$$ Thus the ABS bound is sharpened by the total non-shadow deficit on the top $r$ levels. In the modular setting, we take a coefficient-sensitive viewpoint: the polynomial method depends not just on the degree of the annihilator polynomial $$P_L(t)=\prod_{\ell\in L}(t-\ell)\in\mathbb{F}<em>p[t]$$, but on which binomial terms actually appear in it. This yields a gap-free modular bound depending only on the active support levels of $P_L$. For almost-initial residue patterns $L={0,1,\dots,s-m-1}\cup R \pmod p$ we obtain the collapse $$|\mathcal{F}|\le \sum</em>{i=0}<sup>{m}\binom{n}{s-i}.$$ In particular, for consecutive residues $L={0,1,\dots,s-1}\pmod p$ we get the sharp bound $|\mathcal{F}|\le \binom{n}{s}$, giving a partial negative answer to a question of Alon--Babai--Suzuki: the modular ABS bound $N(n,s,r)$ is not attainable in the consecutive-residue regime whenever $r\ge 2$.