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arxiv: 2606.30186 · v1 · pith:KCTSH7FDnew · submitted 2026-06-29 · 🧮 math.CO · math.NT

Small complete 3-term progression free sets in cyclic groups and vector spaces

Bence Csajb\'ok , Zolt\'an L\'or\'ant Nagy This is my paper

Pith reviewed 2026-06-30 05:22 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords complete 3-AP-free setscyclic groupsvector spaces over finite fieldsarithmetic progressionssaturating setsfinite geometryextremal set theory3-term AP
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The pith

For every m there is a complete 3-AP-free subset of Z_m with size less than 2√m, and the minimal size in F_p^n is at most p^{n/2 + o(n)}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A basic counting argument shows any complete 3-AP-free set in a group of order N must have size at least roughly √N. The paper supplies explicit constructions that meet this lower bound up to a constant factor of 2 in every cyclic group Z_m. For m at least 81 the sets also satisfy the stronger complete (2,-1)-avoiding property; the finitely many smaller cases are checked directly. In the vector-space setting over F_p for fixed odd prime p, the paper proves that the minimal size grows at most like n^{1+ε} times the square root of the space size, for any ε>0. These results show that the square-root order is asymptotically tight in both families of groups.

Core claim

We give explicit constructions of complete 3-AP-free sets in Z_m of size less than 2√m for all m, with the stronger complete (2,-1)-avoiding property for m≥81 and finite verification for m<81. For every fixed odd prime p and ε>0 there exists C_{p,ε} such that a(3-AP, F_p^n) ≤ C_{p,ε} n^{1+ε} p^{n/2} for all n.

What carries the argument

Explicit constructions of complete (2,-1)-avoiding sets in cyclic groups together with asymptotic upper bounds on minimal complete 3-AP-free sets in vector spaces over finite fields.

If this is right

  • The minimal size of a complete 3-AP-free set in Z_m is Θ(√m).
  • The minimal size in F_p^n is p^{n/2 + o(n)}.
  • The square-root lower bound coming from double counting is tight up to a constant or subexponential factor in both settings.
  • Complete 3-AP-free sets provide explicit examples of small saturating sets in the corresponding geometries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same square-root tightness may hold in other families of abelian groups once suitable algebraic constructions are found.
  • The vector-space result suggests that random or algebraic methods can produce near-optimal saturating sets without needing full extremal machinery.
  • The (2,-1)-avoiding strengthening may translate into new bounds on the number of directions or lines determined by the set.

Load-bearing premise

The constructed sets must be free of 3-term arithmetic progressions while remaining maximal, so that every element outside the set completes a 3-AP with two elements inside it.

What would settle it

An m for which every complete 3-AP-free subset of Z_m has size at least 2√m, or an n and odd prime p for which every complete 3-AP-free subset of F_p^n has size larger than C n^{1+ε} p^{n/2} for every constant C.

read the original abstract

A classical extremal problem on progression free sets is to determine the maximum size of a $3$-term arithmetic progression free set in algebraic structures, for instance in intervals of integers or in finite vector spaces. To determine the minimum size of a complete $3$-term arithmetic progression free set is a lower-end analogue of this problem. It is also closely related to complete caps and saturating sets in finite geometry. A simple counting argument shows that the order of magnitude of the minimum size is at least the square root of the cardinality of the structure. Addressing two open problems, we show that this lower bound is essentially tight. First, for every cyclic group $\mathbb{Z}_m$, we give explicit constructions of complete $3$-AP-free sets whose size is less than $2\sqrt m$. For $m\ge81$ the constructed sets satisfy the stronger, so-called complete $(2,-1)$-avoiding property; the remaining cases $m<81$ are covered by a finite verification. Second, we resolve the vector space variant in a weaker sense by showing that for every fixed odd prime $p$ and $\varepsilon>0$, there is a constant $C_{p, \varepsilon}$ such that \[ a(3\text{-}\mathrm{AP},\mathbb{F}_p^n)\le C_{p, \varepsilon}\,n^{1+\varepsilon}\,p^{n/2} =p^{n/2+o(n)} \] holds for the minimum size $a(3\text{-}\mathrm{AP},\mathbb{F}_p^n)$ of a complete 3-AP-free subset of $\mathbb{F}_p^n$, for all $n\ge1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs explicit complete 3-AP-free subsets of every cyclic group Z_m with cardinality strictly less than 2 sqrt(m). For m >= 81 the sets satisfy the stronger complete (2,-1)-avoiding property; the finitely many cases m < 81 are settled by exhaustive verification. In the vector-space setting it proves that for each fixed odd prime p and every epsilon > 0 there exists C = C(p, epsilon) such that the minimal size a(3-AP, F_p^n) satisfies a <= C n^{1+epsilon} p^{n/2} for all n >= 1, i.e., the size is at most p^{n/2 + o(n)}.

Significance. The constructions essentially match the sqrt(|G|) lower bound obtained by double counting, thereby resolving two open questions on the minimal size of complete 3-AP-free sets. The explicit, parameter-free nature of the cyclic constructions together with the finite verification for small m, and the p^{n/2 + o(n)} upper bound in the vector-space case, constitute concrete progress beyond mere existence arguments.

minor comments (3)
  1. [Introduction] The definition of the complete (2,-1)-avoiding property should appear in the introduction or in a dedicated preliminary section before its first use in the cyclic-group constructions.
  2. [Vector spaces] In the vector-space section the dependence of the constant C on p and epsilon should be stated explicitly when the bound is first displayed.
  3. A short table summarizing the sizes obtained for m < 81 would improve readability of the finite-verification claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, detailed summary of our results, and recommendation to accept the manuscript. We are pleased that the constructions and bounds are viewed as resolving the open questions.

Circularity Check

0 steps flagged

No significant circularity; results from explicit constructions and existence arguments

full rationale

The paper establishes its claims via explicit constructions of complete 3-AP-free sets in Z_m (with finite verification for m<81 and a (2,-1)-avoiding property for m≥81) and an existence argument yielding the p^{n/2+o(n)} bound in F_p^n. These are direct constructions and counting-based lower bounds, not self-definitional, fitted predictions, or self-citation chains. No load-bearing step reduces to its own inputs by construction; the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies only on the standard definition of arithmetic progressions in abelian groups and the notion of maximality; no free parameters, new entities, or non-standard axioms appear in the abstract.

axioms (1)
  • standard math Arithmetic progressions are defined via the group operation in the usual way; completeness means maximality under addition of any element.
    Invoked throughout the abstract as the objects under study.

pith-pipeline@v0.9.1-grok · 5849 in / 1199 out tokens · 40088 ms · 2026-06-30T05:22:36.149035+00:00 · methodology

discussion (0)

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Reference graph

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