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

TL;DR

This paper refines intersection theorems by introducing non-shadow and modular support techniques, sharpening bounds on set families with restricted intersections, and providing new insights into the structure of such families.

Contribution

It introduces a multilevel non-shadow refinement of the ABS intersection theorem and a coefficient-sensitive polynomial method for modular bounds, advancing understanding of intersection set families.

Findings

Sharpened bounds on L-intersecting families using non-shadow deficits.

Modular bounds depending on active polynomial support levels.

Counterexamples to attainability of certain modular bounds.

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 $k_i>s-r$ for every $i$, then $|\mathcal{F}| + \sum_{j=s-r+1}^{s} |\mathcal{N}_j(\mathcal{F})| \le N(n,s,r),$ equivalently $|\mathcal{F}| \le \sum_{j=s-r+1}^{s} |\partial_j\mathcal{F}|.$ 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 $P_L(t)=\prod_{\ell\in L}(t-\ell)\in\mathbb{F}_p[t]$, 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 $|\mathcal{F}|\le \sum_{i=0}^{m}\binom{n}{s-i}.$ 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 $r\ge 2$.