Szemerédi regularity lemma
In extremal graph theory, Szemerédi’s regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are regular.
Variants of the lemma use different notions of regularity and apply to other mathematical objects like hypergraphs.
Let X, Y be disjoint subsets of V. The edge density of the pair (X, Y) is defined as:
For every ε > 0 and positive integer m there exists an integer M such that if G is a graph with at least M vertices, there exists an integer k in the range m ≤ k ≤ M and an ε-regular partition of the vertex set of G into k sets.The bound M for the number of parts in the partition of the graph given by the proofs of Szemeredi's regularity lemma is very large, given by a O(ε−5)-level iterated exponential of m. At one time it was hoped that the true bound was much smaller, which would have had several useful applications.
However Gowers (1997) found examples of graphs for which M does indeed grow very fast and is at least as large as a ε−1/16-level iterated exponential of m.[2] We shall find an ε-regular partition for a given graph following an algorithm: We apply a technique called the energy increment argument to show that this process stops after a bounded number of steps.
The graph removal lemma can be used to prove Roth's Theorem on Arithmetic Progressions,[3] and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem.
Szemerédi's regularity lemma does not provide meaningful results in sparse graphs.
Even though the result seems purely theoretical, some attempts [6][7] have been made to use the regularity method as compression technique for large graphs.
[8] This lemma defines a weaker notion of regularity than that of Szemerédi which uses better bounds and can be used in efficient algorithms.
This notion can be extended to graphons by defining a stepping operator.
One of the initial motivations for the development of the weak regularity lemma was the search for an efficient algorithm for estimating the maximum cut in a dense graph.
[8] It has been shown that approximating the max-cut problem beyond 16/17 is NP-hard,[9] however an algorithmic version of the weak regularity lemma gives an efficient algorithm for approximating the max-cut for dense graphs within an
[8] These ideas have been further developed into efficient sampling algorithms for estimating max-cut in dense graphs.
[10] The smaller bounds of the weak regularity lemma allow for efficient algorithms to find an
[11] Graph regularity has further been used in various area of theoretical computer science, such as matrix multiplication[12] and communication complexity.
[5] Intuitively, it provides information between non-regular pairs and could be applied to prove the induced graph removal lemma.
such that the following properties are satisfied: We apply the regularity lemma repeatedly to prove the stronger version.
where the following properties are satisfied: The corollary explores deeper the small energy increment.
It gives us a partition together with subsets with large sizes from each part, which are pairwise regular.
The full version can be proved by picking more subsets from each part that are mostly pairwise regular and combine them together.
We apply the strong regularity lemma to find equitable
[15] Extensions of the regularity method to hypergraphs were obtained by Rödl and his collaborators[16][17][18] and Gowers.
[19][20] János Komlós, Gábor Sárközy and Endre Szemerédi later (in 1997) proved in the blow-up lemma[21][22] that the regular pairs in Szemerédi regularity lemma behave like complete bipartite graphs under the correct conditions.
The first constructive version was provided by Alon, Duke, Lefmann, Rödl and Yuster.
[23] Subsequently, Frieze and Kannan gave a different version and extended it to hypergraphs.
[24] They later produced a different construction due to Alan Frieze and Ravi Kannan that uses singular values of matrices.
[25] One can find more efficient non-deterministic algorithms, as formally detailed in Terence Tao's blog[26] and implicitly mentioned in various papers.
[30] Terence Tao has also provided a proof of the lemma based on spectral theory, using the adjacency matrices of graphs.
-regular partition to require that the vertex sets all have the same size, while collecting the leftover vertices in an "error"-set
Szemerédi's regularity lemma can be interpreted as saying that the space of all graphs is totally bounded (and hence precompact) in a suitable metric (the cut distance).
