Forbidden subgraphs in divisor graphs and an Erdős divisibility problem

Abstract: Erdős asked for the largest size $f(n)$ of a subset of ${1,\dots,n}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such subsets satisfies $q(n)=β_2^{n+o(n)}$ for a computable constant $β_2$. To prove this, we recast the divisibility constraint as forbidding a certain directed subgraph in the divisor graph on ${1,\dots,n}$ and prove a more general result: for any finite family of connected forbidden subgraphs of the divisor graph, both the extremal density and counting rate are effectively computable. The proof uses a theorem of McNew on local statistics of divisor graphs.

• The paper presents the asymptotic evaluation of the maximal subset density, f(n) = c2·n + o(n), sharpening bounds for the Erdős two-fork problem.
• It leverages local divisor graph block analysis to compute effective constants, establishing computable bounds for both extremal density and exponential growth rates.
• The unified framework addresses diverse divisibility constraints and has implications for algorithmic sampling and extremal combinatorial optimization.

Forbidden Subgraphs in Divisor Graphs and the Erdős Two-Fork Problem

Introduction and Problem Statement

This paper addresses an extremal problem in additive number theory initially posed by Erdős, namely: for a subset $A \subseteq \{1, \ldots, n\}$ such that no element of $A$ divides two others, what is the asymptotic density of the largest such subset, $f(n)$, as $n \to \infty$? Additionally, what can be said about the number $q(n)$ of such subsets?

The approach leverages the structure of divisor graphs---graphs with vertex set $\{1, \ldots, n\}$ and a directed edge $u \to v$ whenever $u\mid v$---to reinterpret constraints like “no element divides two others” as forbidden subgraph patterns. In this language, the problem reduces to a Turán-type extremal question, forbidding the directed two-fork (a root with directed edges to two leaves) as an induced pattern.

Main Results

Extremal Density and Counting

The central result is the asymptotic evaluation of $f(n)$ and $q(n)$ in terms of explicit, effectively computable constants:

• There exists a computable $A$0 such that $A$1. The constant $A$2 represents the extremal density of two-fork-free sets.
• There is a computable $A$3 such that $A$4, describing the exponential growth rate of two-fork-free sets.

The constant $A$5 is explicitly estimated to satisfy $A$6, refining previous bounds. Notably, the irrationality of $A$7, as originally questioned by Erdős, remains unresolved.

Generalized Forbidden Subgraph Framework

The results are formulated within a broader framework: for any finite family $A$8 of connected (directed or undirected) forbidden subgraphs of the divisor graph, the maximal density $A$9 and the exponential growth constant $f(n)$0 for $f(n)$1-free subsets are both effectively computable. This unifies a variety of problems---e.g., $f(n)$2-forks, $f(n)$3-chains, in-forks, and divisor forests---as special cases within a single analytic framework.

The proof leverages a structural theorem of McNew, which expresses statistics of local divisor graph configurations as convergent series over block types. The relevant class of “admissible” properties is those that are downward closed, respect coprime decomposability, and are invariant under scaling.

Effective Computability and Truncation Analysis

The local contributions to the constants ($f(n)$4, $f(n)$5) come from evaluating differences over specific rooted divisor-graph blocks, weighted by an explicit measure. The computation is rendered feasible via effective enumeration and exhaustive search or combinatorial optimization on small instances (orbits under scaling and coprimality), ensuring convergence and bounding truncation errors.

For the two-fork case:

• With sufficiently deep enumeration (e.g., 32,660 rooted components), one obtains $f(n)$6.
• For the counting rate, $f(n)$7 is established.

Structural and Theoretical Implications

This work rigorously connects divisor-graph extremal problems with classical Turán-type vertex problems, enabling tools from extremal combinatorics to address arithmetically-motivated subset problems. The downward-closed, coprime-decomposable, scale-invariant framework for $f(n)$8-avoidance captures a wide range of natural divisibility constraints. By showing that for any such finite family, the extremal density and growth rates are computable, the paper establishes a blueprint for systematic classification of forbidden pattern problems in divisor graphs.

Practically, the results yield precise bounds and computational methods for longstanding extremal problems in the structure of integer subsets closed under divisibility constraints. The theoretical implication is a unification of previously disparate results under a single analytic and combinatorial umbrella.

Future Directions

Key open questions include the irrationality of $f(n)$9, the improvement of numerical bounds (which is currently bottlenecked by the computational complexity of block enumeration), and extensions to infinite families of forbidden subgraphs (e.g., divisor forests). The framework may be leveraged for algorithmic generation and sampling of extremal or random $n \to \infty$0-free sets, and for analyzing analogous properties in multiplicative number theory.

There is also potential to apply similar local block analysis to related extremal problems on other algebraic graphs or posets, as well as to study stability and structure theorems for near-maximum density sets.

Conclusion

The paper establishes the computability of extremal density and counting exponents for subsets of $n \to \infty$1 avoiding prescribed connected patterns in the divisor graph. In particular, it gives refined estimates for the classic Erdős two-fork problem, providing both improved bounds and a broader analytic framework applicable to a variety of divisibility-based graph constraints. The connection drawn between arithmetic extremal questions and vertex Turán problems in orientation-labeled graphs deepens the interface between additive combinatorics, extremal graph theory, and number theory.

Reference: "Forbidden subgraphs in divisor graphs and an Erdős divisibility problem" (2604.17613)