• Thumbnail for 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...
    39 KB (6,164 words) - 06:47, 12 May 2025
  • In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set X {\displaystyle X} ,...
    8 KB (1,116 words) - 21:29, 26 January 2025
  • Regular (redirect from Regularity)
    of Regularity, also called the Axiom of Foundation, an axiom of set theory asserting the non-existence of certain infinite chains of sets Partition regularity...
    8 KB (1,019 words) - 01:20, 25 May 2025
  • pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order n; that is, there...
    9 KB (1,144 words) - 21:57, 21 May 2025
  • the random-like parts. This is an extension of Szemerédi's regularity lemma that partitions any given graph into bounded number parts such that edges between...
    18 KB (3,386 words) - 02:09, 23 September 2024
  • Thumbnail for Julian Sahasrabudhe
    b {\displaystyle a,b,a^{b}} is monochromatic, demonstrating the partition regularity of complex exponential patterns. This work marks a crucial development...
    13 KB (1,319 words) - 00:58, 26 March 2025
  • Thumbnail for Extremal graph theory
    simplest forms, the graph counting lemma uses regularity between pairs of parts in a regular partition to approximate the number of subgraphs, and the...
    10 KB (1,360 words) - 09:43, 1 August 2022
  • Thumbnail for Fermat's right triangle theorem
    on 2013-01-20 Cooper, Joshua; Poirel, Chris (2008), Pythagorean partition-regularity and ordered triple systems with the sum property, arXiv:0809.3478...
    16 KB (2,279 words) - 04:11, 14 May 2025
  • algorithmic version of the Szemerédi regularity lemma to find an ϵ {\displaystyle \epsilon } -regular partition. Lemma 1: Fix k and γ {\displaystyle \gamma...
    8 KB (1,274 words) - 09:51, 15 March 2025
  • {N} } , the Stone–Čech compactification of the natural numbers. Partition regularity: if S {\displaystyle S} is piecewise syndetic and S = C 1 ∪ C 2 ∪...
    4 KB (465 words) - 12:46, 8 November 2024
  • "Extremal Sidon Sets are Fourier Uniform, with Applications to Partition Regularity". Journal de théorie des nombres de Bordeaux. 35 (1): 115–134. arXiv:2110...
    17 KB (2,208 words) - 02:49, 14 April 2025
  • Thumbnail for Axiom of choice
    theorem. The axiom of choice is equivalent to the statement that every partition has a transversal. In many cases, a set created by choosing elements can...
    59 KB (7,917 words) - 15:47, 15 May 2025
  • {\displaystyle E} as an instrument to split A {\displaystyle A} into two partitions: the part of A {\displaystyle A} which intersects with E {\displaystyle...
    19 KB (2,952 words) - 10:25, 9 April 2025
  • preregularity, rather than regularity, that matters in these situations. However, definitions are usually still phrased in terms of regularity, since this condition...
    16 KB (2,156 words) - 02:43, 25 March 2025
  • G_{l}^{(2)}} via a partition of the vertex set. As a result, we have the total data of hypergraph regularity as follows: a partition of E ( K n ) {\displaystyle...
    14 KB (2,349 words) - 22:49, 27 February 2025
  • Thumbnail for Finite geometry
    mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite...
    22 KB (2,841 words) - 13:36, 12 April 2024
  • Thumbnail for Half graph
    Therefore, it is not possible to strengthen the regularity lemma to show the existence of a partition for which all pairs are regular. On the other hand...
    9 KB (1,203 words) - 02:38, 29 July 2024
  • Thumbnail for Neil Hindman
    \mathbb {N} } . This theorem highlights the relationship between the partition regularity of the natural numbers and ultrafilters, offering a fundamental result...
    8 KB (720 words) - 04:12, 28 May 2024
  • schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by John von Neumann), to Zermelo set theory yields the...
    46 KB (6,252 words) - 13:43, 7 June 2025
  • Thumbnail for Equivalence relation
    {\displaystyle a=c} (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements...
    31 KB (4,473 words) - 10:22, 23 May 2025
  • comprehension, or the axiom of regularity and axiom of pairing. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any...
    10 KB (1,327 words) - 06:43, 21 May 2024
  • are partitioned into finitely many subsets, there exist arbitrarily large sets of numbers all of whose sums belong to the same subset of the partition. The...
    6 KB (780 words) - 14:08, 14 January 2024
  • Thumbnail for Graphon
    {\mathcal {P}}} . The statement that a graph G {\displaystyle G} has a regularity partition is equivalent to saying that its associated graphon W G {\displaystyle...
    33 KB (5,460 words) - 19:20, 21 February 2025
  • law of excluded middle), all instances of regularity hold. In a context with an axiom of separation, regularity also implies excluded middle (for the predicates...
    24 KB (4,195 words) - 22:44, 26 March 2025
  • Thumbnail for Smoothness
    given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the...
    25 KB (3,930 words) - 22:46, 20 March 2025
  • monochromatic, may be seen as a special case of Rado's theorem concerning the regularity of the system of equations x T = ∑ i ∈ T x { i } , {\displaystyle x_{T}=\sum...
    4 KB (606 words) - 03:13, 12 March 2024
  • be the energy function defined in Szemerédi regularity lemma. Essentially, we can find a pair of partitions P , Q {\displaystyle {\mathcal {P}},{\mathcal...
    32 KB (5,077 words) - 23:54, 9 March 2025
  • quasi-conformal...), a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity. Manifold with boundary Manifold...
    7 KB (869 words) - 08:54, 6 December 2024
  • Thumbnail for Rook's graph
    of the largest independent set is equal to the number of cliques in a partition of the graph's vertices into a minimum number of cliques. In a rook's...
    31 KB (3,777 words) - 20:42, 16 December 2024
  • candidate u to the data f. The parameter γ > 0 controls the tradeoff between regularity and data fidelity. There are fast algorithms for the exact minimization...
    23 KB (3,645 words) - 09:43, 26 February 2025