The Symmetry Preserving Removal Lemma

In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemer\'edi's Theorem on arithmetic progressions. If in an Abelian group $A$ there are sets $S_1,S_2...,S_t$ such that the number of arithmetic progressions $x_1,x_2,...,x_t$ with $x_i\in S_i$ is $o(|A|^2)$ then we can shrink each $S_i$ by $o(|A|)$ elements such that the new sets don't have such a diagonal arithmetic progression.


Introduction
A directed k-uniform hyper-graph H on the vertex set V is a subset of V k such that there is no repetition in the k coordinates. A homomorphism between two directed k-uniform hyper-graphs F and H with vertex sets V (F ) and V (H) is a map f : V (F ) → V (H) such that (f (a 1 ), f (a 2 ), . . . f (a k )) is in H whenever (a 1 , a 2 , . . . a k ) is in F . The automorphism group Aut(H) is the group of bijective homomorphisms π : V (H) → V (H). The homomorphism density t(F, G) of F in G is the probability that a random map f : V (G) → V (H) is a homomorphism.
The so-called hyper-graph removal lemma ( [3], [4], [1], [2], [7])(in the directed setting) says the following Using this deep result we observe that the edge removal can be done in a way that the symmetries of G remain preserved.
Then for every fixed e ∈ F and for random element π ∈ Aut(G) the probability that π(f (e)) ∈ G \ S is less that 1/|F | and so there is some π ∈ Aut(G) with π(f (F )) ⊆ G \ S which is contradiction.
The argument given for the symmetry preserving removal is very general. It applies for various modified versions of the removal lemma. An important such version is the t-partite removal lemma where t is a fixed natural number.
An automorphism is a bijective homomorphism from G 1 to G 1 and the homomorphism density t(G 1 , G 2 ) is the probability that a random t tuple of maps We give an example for an application of the symmetry preserving removal lemma and then we generalize it in the next chapter.  [9] for Abelian groups and generalized for groups by Kral, Serra and Vena [8].

Cayley Hypergraphs
In this chapter we describe a potential way of generalizing Cayley graphs to the hypergraphs setting and then discuss the symmetry preserving removal lemma on such graphs.

is called a Cayley hypergraph if its automorphism group contains H with the previous action.
This definition is very general so we will start to analyze a special setting. Assume that all the groups G 1 , G 2 , . . . , G t are isomorphic to an Abelian group A. Furthermore, to get something interesting we want to assume that H is not too big and not too small. Let C = {C 1 , C 2 , . . . , C r } be a collection of k-element subsets of {1, 2, . . . , t}. Each set C i defines a projection p i : H → A k to the coordinates in C i . Assume that the factor group A C i /p i (H) ∼ = A and let ψ i : A C i → A be a homomorphism with kernel p i (H). Now we pick a subsets S i ⊆ A for 1 ≤ i ≤ r and we define the graph where ψ −1 i (S i ) is the union of cosets in A C i of p i (H) representing an element in S i . Note that the way we produced H k,t (A, {S i }, C) guarantees that its automorphism group contains H as a subgroup.
The symmetry preserving removal lemma for t-partite hypergraphs directly implies the following lemma: Example 2.: This example uses an idea by Solymosi [6] who showed that the Hypergraph Removal Lemma implies Szemere'di's theorem on arithmetic progressions (even in a multi dimensional setting). Let t be a natural number, k = t−1 and A be an Abelian group. We define H to be the subgroup in A t of the elements (a 1 , a 2 The functions ψ i are computed in a way that ker(ψ −1 i ) is the projection of H to the coordinates in C i .
Let F be the complete t partite t − 1 uniform hypergraph on four point. Lemma 2.1 applied to F and the above hypergraph H t−1,t (A, {S i }, C) implies that if the system has o(|A| t ) solutions then we can delete o(|A|) elements from each S i such that the previous system has no solution. It is clear that x 1 , x 2 , . . . , x t are forming a t term arithmetic progression and in fact any such progression with x i ∈ S i gives rise to |A| t−2 solution of the previous system. Using this we obtain the following: Theorem 3 (Diagonal Szemerédi Theorem) For every ǫ > 0 there exists a δ > 0 such that if A is an Abelian group, S 1 , S 2 , . . . , S t are subsets in A and there are at most δ|A| 2 t-tuples x 1 , x 2 , . . . , x t with x i ∈ S i such that they are forming a t term arithmetic progression then we can shrink each S i by at most ǫ|A| elements such that the new sets don't have such a configuration.
This theorem implies Szemerédi's theorem [10] if we apply it for S = S i , 1 ≤ i ≤ t since S contains the trivial progressions a, a, a . . . , a which are only removable if we delete the whole set S.