An Erdős--Trotter problem on antichains with multiplicity $r$ on each occurring level

TL;DR

The paper resolves the Erdős–Trotter problem for $r$-multiplicity antichains by determining the threshold $n_0(r)$ at which one can realize $n-3$ distinct set sizes in all large instances. It proves the exact values $n_0(2)=3$ and $n_0(3)=8$, and establishes sharp two-sided bounds $2r+2 \le n_0(r) \le 2r+2\log_2 r + O(\log_2\log_2 r)$ for all $r\ge4$, with a separate constructive argument showing $g(n,r)=n-3$ for large $n$ and a universal obstruction ensuring $g(n,r)\le n-3$. The approach blends classical extremal-set-theory tools (central-binomial bounds, star/triangle classifications) with a novel label-antichain gadget and a careful level-by-level construction, yielding explicit near-optimal constructions and enabling exact values for small $r$. These results advance the Erdős–Trotter program by clarifying the asymptotic behavior of the threshold and by providing concrete constructions that nearly match the universal obstruction, offering a foundation for future tightening of the remaining logarithmic terms.

Abstract

Fix an integer $r\ge2$. For each $n$ we consider families $\mathcal F\subseteq 2^{[n]}$ that form an antichain and have the property that, for every $t$, if there exists $A\in\mathcal F$ with $|A|=t$ then there exist at least $r$ members of $\mathcal F$ of size $t$. A problem of Erdős and Trotter asserts that, for each fixed $r$, there exists a threshold $n_0(r)$ such that whenever $n>n_0(r)$ one can achieve $n-3$ distinct set sizes in such a family, and asks for estimates on $n_0(r)$. We compute that $n_0(2)=3$ and $n_0(3)=8$. For all $r\ge4$ we prove matching linear bounds up to lower-order terms, namely $$ 2r+2 \le n_0(r) \le 2r+2\log_2 r + O(\log_2\log_2 r). $$

Theorems & Definitions (40)

• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof