[Paper Review] A connection between matchings and removal in abelian groups

James Aaronson|arXiv (Cornell University)|Dec 13, 2016

Limits and Structures in Graph Theory7 references3 citations

TL;DR

This paper establishes a connection between the maximum size of additive matchings (tricolored sum-free sets) and bounds for Green's arithmetic removal lemma in finite abelian groups, particularly cyclic groups. By adapting techniques from Fox and Lovász, it shows that polynomial or subexponential bounds on additive matchings imply corresponding subexponential bounds on the removal lemma, with explicit quantitative trade-offs involving functions f and g that govern matching density and removal efficiency.

ABSTRACT

In a finite abelian group $G$, define an additive matching to be a collection of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i = j = k$. In the case that $G = \mathbb{F}_2^n$, Kleinberg, building on work of Croot-Lev-Pach and Ellenberg-Gijswijt, proved a polynomial upper bound on the size of an additive matching. Fox and Lovász used this to deduce polynomial bounds on Green's arithmetic removal lemma in $\mathbb{F}_2^n$. If $G$ is taken to be an arbitrary finite abelian group, the questions of bounding the size of an additive matching and giving bounds for Green's arithmetic removal lemma are much less well understood. In this note, we adapt the methods of Fox and Lovász to prove that, provided we can assume a sufficiently strong bound on the size of an additive matching in cyclic groups, a similar bound should hold in the case of removal.

Motivation & Objective

• To establish a quantitative link between the maximum density of additive matchings in cyclic groups and the bounds for Green's arithmetic removal lemma.
• To extend the polynomial bounds on removal in F₂ⁿ to general finite abelian groups, particularly Z/NZ, using matching size constraints.
• To formalize conditions under which subexponential removal bounds can be achieved, based on the decay rate of additive matching densities.
• To generalize the Fox-Lovász method from prime-order groups to arbitrary cyclic groups via reduction and structural analysis.
• To provide a converse relationship showing that removal bounds imply matching size bounds, confirming the tightness of the connection.

Proposed method

• Adapts the Fourier-analytic and combinatorial framework of Fox and Lovász to cyclic groups, using additive matching density as a key parameter.
• Employs a greedy triangle-packing argument to identify a large set of disjoint triangles in the input sets A, B, C, ensuring that removal of fewer than εN elements cannot eliminate all triangles.
• Reduces the problem from general Z/NZ to prime-order groups by selecting a prime N ≈ 2M and using the reduction map modulo N, preserving triangle structures.
• Applies Lemma 2.1 to bound the size of sets X, Y, Z based on the number of triangles they form and the density of additive matchings in the group.
• Uses a function f(N) to bound the maximum density of additive matchings in Z/NZ, and a function g(ρ) to control the trade-off between δ (triangle density) and ε (removal cost).
• Derives the key inequality ε ≲ g(δ)√f(1/(g(δ)δ)) by combining the greedy triangle packing with the matching density assumption and monotonicity conditions on f and g.

Experimental results

Research questions

• RQ1 Under what conditions on the density of additive matchings in Z/NZ can we deduce strong bounds for Green’s arithmetic removal lemma?
• RQ2 How does the decay rate of the maximum matching density f(N) influence the best possible removal bound ε in terms of δ?
• RQ3 Can the Fox-Lovász method for F_p^n be extended to arbitrary finite abelian groups, particularly cyclic groups?
• RQ4 What is the quantitative relationship between additive matching size and the efficiency of triangle removal in abelian groups?
• RQ5 Is there a converse relationship where removal bounds imply bounds on additive matching size?

Key findings

• The paper proves that if the maximum density of an additive matching in Z/NZ is bounded by f(N), then the removal lemma bound satisfies ε ≲ g(δ)√f(1/(g(δ)δ)) for an appropriate function g.
• Under a Behrend-type bound f(N) ≲ exp(−c√log N), the paper shows ε ≲ exp(−c₁√log(1/δ)), a subexponential improvement over tower-type bounds.
• For a weaker assumption f(N) ≲ log⁻²⁻ᵞN with γ > 0, the bound becomes ε ≲ log(1/δ)⁻ᴼ⁽¹⁾, indicating a polynomial-type removal cost.
• The convergence condition ∑ᵢ 1/g(2⁻ⁱ) ≤ 1/4 is necessary to ensure the iterative argument in the proof holds, and failure of this implies no nontrivial bound can be derived.
• The method generalizes from prime-order groups to arbitrary cyclic groups via reduction modulo a prime N ≈ 2M, preserving triangle structures and enabling the extension of results.
• A converse is established: if removal bounds imply ε ≲ f(1/δ), then additive matchings cannot exceed density f(N), confirming the tightness of the connection.