[Paper Review] On the removal lemma for linear configurations in finite abelian groups

Lluís Vena|arXiv (Cornell University)|May 9, 2015

Limits and Structures in Graph Theory18 references3 citations

TL;DR

This paper introduces a hypergraph-based representation framework to generalize combinatorial removal lemmas to linear configurations in finite abelian groups. By modeling systems of equations as hypergraphs, the authors establish a removal lemma for homomorphisms between finite abelian groups, proving that configurations can be eliminated by removing few elements under a coprime determinantal condition, with applications to arithmetic progressions and permutation patterns.

ABSTRACT

We present a general framework to represent discrete configuration systems using hypergraphs. This representation allows one to transfer combinatorial removal lemmas to their analogues for configuration systems. These removal lemmas claim that a system without many configurations can be made configuration-free by removing a few of its constituent elements. As applications of this approach we give, an alternative proof of the removal lemma for permutations by Cooper, a general version of a removal lemma for linear systems in finite abelian groups, an interpretation of the mentioned removal lemma in terms of subgroups, and an alternative proof of the counting version of the multidimensional Szemerédi theorem in abelian groups with generalizations.

Motivation & Objective

• To develop a general framework for representing discrete configuration systems using hypergraphs.
• To extend combinatorial removal lemmas from graphs and hypergraphs to systems of linear equations in finite abelian groups.
• To establish a removal lemma for homomorphism systems in finite abelian groups under a coprime determinantal condition.
• To provide alternative proofs for known results, such as the removal lemma for permutations and the multidimensional Szemerédi theorem.
• To connect discrete combinatorial structures with algebraic group-theoretic configurations via representable hypergraph systems.

Proposed method

• Represent linear configuration systems as hypergraphs, where vertices correspond to group elements and edges represent solution tuples.
• Use the concept of representability to link hypergraph configurations to systems of equations over finite abelian groups.
• Apply the hypergraph removal lemma (Theorem 11) to the constructed hypergraph representations to derive removal results.
• Reduce the problem of removing solutions to linear systems to removing edges in a hypergraph, leveraging the regularity and counting properties of hypergraph models.
• Utilize the structure of homomorphisms between finite abelian groups to define and analyze representable systems, especially through cluster partitioning and edge coloring.
• Introduce a transformation that isolates non-trivial variables and reduces the system to a core hypergraph representation, preserving solution counts up to a constant factor.

Experimental results

Research questions

• RQ1 Can combinatorial removal lemmas be generalized from hypergraphs to systems of linear equations in finite abelian groups?
• RQ2 Under what algebraic conditions on the defining matrix of a linear system does a removal lemma hold in finite abelian groups?
• RQ3 Can the removal lemma for permutations be recovered through a hypergraph representation framework?
• RQ4 Is there a hypergraph-theoretic interpretation of the multidimensional Szemerédi theorem in abelian groups?
• RQ5 How can the structure of homomorphisms between finite abelian groups be encoded and analyzed via representable hypergraph systems?

Key findings

• A removal lemma is established for systems of linear equations in finite abelian groups when the maximal determinantal of the coefficient matrix is coprime to the group order.
• The number of solutions to a system of equations can be made zero by removing at most ε|G| elements from each variable set, provided the number of solutions is less than δ|G|^{m-1} for a suitable δ.
• The framework provides an alternative proof of the removal lemma for permutations, recovering Cooper’s result via hypergraph representation.
• The method yields a new proof of the counting version of the multidimensional Szemerédi theorem in abelian groups, with generalizations to arbitrary finite abelian groups.
• The hypergraph representation preserves solution counts up to a constant factor, enabling transfer of removal lemmas from hypergraphs to algebraic configuration systems.
• The construction allows for the isolation of non-trivial variables and reduction to a core hypergraph system, maintaining the solution structure under removal operations.