Hypergraph regularity method: Difference between revisions

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* <math> \mathcal{K}_j(\mathcal{G}^{(i)}) </math> is the family of all <math> j </math>-element vertex sets that span the clique <math> K_j^{(i)} </math> in <math> \mathcal{G}^{(i)} </math>. In particular, <math> \mathcal{K}_j(\mathcal{G}^{(1)}) = K_l^{(j)}(V_1, \ldots, V_l) </math> is a complete <math> l </math>-partite <math> j </math>-graph

Suppose<blockquote>'''Definition [Relative density].''' For <math> j \delta_j,geq d_j3 </math>, arefix positivesome real numbers andclasses <math> rV_{i_1}, \geqldots, 1V_{i_j} </math> is an integer.of <math> \mathcal{G}^{(j1)} </math> iswith (<math> 1 \delta_j,leq d_j, ri_1 </math>)-regular with respect to\ldots <math> i_j \mathcal{G}^{(j-1)}leq l </math>. if for any choice of classesSuppose <math> V_{i_1},r \ldots,geq V_{i_j}1 </math> andis anyan collection ofinteger. subhypergraphsLet <math> \mathbf{Q}^{(j-1)} = \{ Q_1^{(j-1)}, \ldots, Q_r^{(j-1)} \} </math> be a subhypergraph of the induced <math> j </math>-partite graph <math> \mathcal{G}^{(j-1)}[V_{i_1}, \ldots, V_{i_j}] </math>. Define the relative satisfyingdensity

<blockquote>'''Definition [Relative density]'''

For <math>d\left(\mathcal{G}^{(j)} \vert \mathbf{Q}^{(j-1)}\right) = \geqfrac{\left|\mathcal{G}^{(j)} 3\cap \cup_{s \in [r]} \mathcal{K}_j(Q_s^{j-1})\right|}{\left|\cup_{s \in [r]} \mathcal{K}_j(Q_s^{j-1})\right|}</math></blockquote>In what follows is the appropriate notion of pseudorandomness that the regularity method will use. Informally, fixby somethis classesconcept of regularity, <math> V_{i_1},(j-1) \ldots,</math>-edges V_(<math> \mathcal{i_jG}^{(j-1)} </math>) ofhave some control over <math> j </math>-edges (<math> \mathcal{G}^{(1j)} </math>). withFormally,<blockquote>'''Definition [(<math> 1 \leqdelta_j, i_1d_j, r </math>)-regularity].''' \ldotsSuppose < i_jmath> \leqdelta_j, ld_j </math>. Supposeare positive real numbers and <math> r \geq 1 </math> is an integer. Let<math> \mathcal{G}^{(j)} </math> is (<math> \delta_j, d_j, r </math>)-regular with respect to <math> \mathcal{G}^{(j-1)} </math> if for any choice of classes <math> V_{i_1}, \ldots, V_{i_j} </math> and any collection of subhypergraphs <math> \mathbf{Q}^{(j-1)} = \{ Q_1^{(j-1)}, \ldots, Q_r^{(j-1)} \} </math> be a subhypergraph of the induced <math> j </math>-partite graph <math> \mathcal{G}^{(j-1)}[V_{i_1}, \ldots, V_{i_j}] </math>. Define the relative density satisfying

<math>d\left(\mathcal{G}^{(j)} \vert \mathbf{Q}^{(j-1)}\right) = \frac{\left|\mathcal{G}^{(j)} \cap \cup_{s \in [r]} \mathcal{K}_j(Q_s^{j-1})\right|}{\left|\cup_{s \in [r]} \mathcal{K}_j(Q_s^{j-1})\right|}</math></blockquote>The following is the appropriate notion of pseudorandomness that the regularity method will use. Informally, by this concept of regularity, <math> j-1 </math>-edges (<math> \mathcal{G}^{(j-1)} </math>) have some control over <math> j </math>-edges (<math> \mathcal{G}^{(j)} </math>).<blockquote>'''Definition [(<math> \delta_j, d_j, r </math>)-regularity]'''

Suppose <math> \delta_j, d_j </math> are positive real numbers and <math> r \geq 1 </math> is an integer. <math> \mathcal{G}^{(j)} </math> is (<math> \delta_j, d_j, r </math>)-regular with respect to <math> \mathcal{G}^{(j-1)} </math> if for any choice of classes <math> V_{i_1}, \ldots, V_{i_j} </math> and any collection of subhypergraphs <math> \mathbf{Q}^{(j-1)} = \{ Q_1^{(j-1)}, \ldots, Q_r^{(j-1)} \} </math> of <math> \mathcal{G}^{(j-1)}[V_{i_1}, \ldots, V_{i_j}] </math> satisfying

<math>\left|\cup_{s \in [r]}\mathcal{K}_j(Q_s)^{(j-1)}\right| \geq \delta_j \left|\mathcal{K}_j(\mathcal{G}^{(j-1)}[V_{i_1}, \ldots, V_{i_j}])\right|</math>

we have <math> d(\mathcal{G}^{(j)} \vert \mathbf{Q}^{(j-1)}) = d_j \pm \delta_j </math></blockquote>Roughly speaking, following describes the pseudorandom blocks into which the hypergraph regularity lemma decomposes any large enough hypergraph. In Szemerédi regularity 2-edges are regularized versus 1-edges (vertices). In this generalized notion, <math> j </math>-edges are regularized versus <math> (j-1) </math>-edges, for all <math> 2\leq j\leq h </math>.<blockquote>'''Definition [<math> (\delta, \mathbf{d},r) </math>-regular <math> (l, h) </math>-complex].''' An <math> (l,h) </math>-complex <math> \mathbf{G} </math> is a system <math> \{\mathcal{G}^{(j)}\}_{j=1}^h </math> of <math> l </math>-partite <math> j </math> graphs <math> \mathcal{G}^{(j)} </math> satisfying <math> \mathcal{G}^{(j)} \subset \mathcal{K}_j(\mathcal{G}^{(j-1)}) </math>. Given vectors of positive real numbers <math> \delta = (\delta_2, \ldots, \delta_h) </math>, <math> \mathbf{d} = (d_2, \ldots, d_h) </math>, and an integer <math> r \geq 1 </math>, we say <math> (l, h) </math>-complex is <math> (\delta, \mathbf{d},r) </math>-regular if

An <math> (l,h) </math>-complex <math> \mathbf{G} </math> is a system <math> \{\mathcal{G}^{(j)}\}_{j=1}^h </math> of <math> l </math>-partite <math> j </math> graphs <math> \mathcal{G}^{(j)} </math> satisfying <math> \mathcal{G}^{(j)} \subset \mathcal{K}_j(\mathcal{G}^{(j-1)}) </math>. Given vectors of positive real numbers <math> \delta = (\delta_2, \ldots, \delta_h) </math>, <math> \mathbf{d} = (d_2, \ldots, d_h) </math>, and an integer <math> r \geq 1 </math>, we say <math> (l, h) </math>-complex is <math> (\delta, \mathbf{d},r) </math>-regular if

* For each <math> 1 \leq i_1 < i_2 \leq l </math>, <math> \mathcal{G}^{(2)}[V_{i_1},V_{i_2}] </math> is <math> \delta_2 </math>-regular with density <math> d_2 \pm \delta_2 </math>.

* For each <math> 3 \leq j \leq h </math>, <math> \mathcal{G}^{(j)} </math> is (<math> \delta_j, d_j, r </math>)-regular with respect to <math> \mathcal{G}^{(j-1)} </math>

</blockquote>The following describes the equitable partition that the hypergraph regularity lemma will induce. The <math> (\mu, \delta, \mathbf{d}, r) </math>-equitable family of partition is a sequence of partitions of 1-edges (vertices), 2-edges (pairs), 3-edges (triples), etc. <blockquote>'''Definition [<math> (\mu, \delta, \mathbf{d}, r) </math>-equitable partition].''' Let <math> \mu > 0 </math> be a real number, <math> r \geq 1 </math> be an integer, and <math> \delta = (\delta_2, \ldots, \delta_{k-1}) </math>, <math> \mathbf{d}=(d_2, \ldots, d_{k-1}) </math> be vectors of positive reals. Let <math> \mathbf{a} = (a_1, \ldots, a_{k-1}) </math> be a vector of positive integers and <math> V </math> be an <math> n </math>-element vertex set. We say that a family of partitions <math> \mathcal{P} = \mathcal{P}(k-1, \mathbf{a}) = \{\mathcal{P}^{(1)}\, \ldots, \mathcal{P}^{(k-1)}\} </math> on <math> V </math> is <math> (\mu, \delta, \mathbf{d}, r) </math>-equitable if it satisfies the following

Let <math> \mu > 0 </math> be a real number, <math> r \geq 1 </math> be an integer, and <math> \delta = (\delta_2, \ldots, \delta_{k-1}) </math>, <math> \mathbf{d}=(d_2, \ldots, d_{k-1}) </math> be vectors of positive reals. Let <math> \mathbf{a} = (a_1, \ldots, a_{k-1}) </math> be a vector of positive integers and <math> V </math> be an <math> n </math>-element vertex set. We say that a family of partitions <math> \mathcal{P} = \mathcal{P}(k-1, \mathbf{a}) = \{\mathcal{P}^{(1)}\, \ldots, \mathcal{P}^{(k-1)}\} </math> on <math> V </math> is <math> (\mu, \delta, \mathbf{d}, r) </math>-equitable if it satisfies the following

* <math> \mathcal{P}^{(1)} = \{V_i \colon i \in [a_1]\} </math> is equitable vertex partition of <math> V </math>. That is <math> |V_1| \leq \ldots \leq |V_{a_1}| \leq | V_1| + 1 </math>

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* For all but at most <math> \mu n^k </math> <math> k </math>-tuples <math> K \in \binom{V}{k} </math> there is unique <math> (\delta, \mathbf{d},r) </math>-regular <math> (k, k-1) </math>-complex <math> \mathbf{P} = \{P^{(j)}\}_{j=1}^{k-1} </math> such that <math> P^{(j)} </math> has as members <math> \binom{k}{j} </math> different partition classes from <math> \mathcal{P}^{(j)} </math> and <math> K \in \mathcal{K}_k(P^{(k-1)}) \subset \ldots \subset \mathcal{K}_k(P^{(1)}) </math>

'''Definition [Regularity with respect to a partition].''' We say that a <math> k </math>-graph <math> \mathcal{H}^{(k)} </math> is <math> (\delta_k, r) </math>-regular with respect to a family of partitions <math> \mathcal{P} </math> if all but at most <math> \delta_k n^k </math> <math> k- </math>edges <math> K </math> of <math> \mathcal{H}^{(k)} </math> have the property that <math> K \in \mathcal{K}_k(\mathcal{G}^{(1)}) </math> and if <math> \mathbf{P} = \{P^{(j)}\}_{j=1}^{k-1} </math> is unique <math> (k, k-1) </math>-complex for which <math> K \in \mathcal{K}_k(P^{(k-1)}) </math>, then <math> \mathcal{H}^{(k)} </math> is <math> (\delta_k, d(\mathcal{H}^{(k)} \vert P^{(k-1)}), r) </math> regular with respect to <math> \mathcal{P} </math>.</blockquote>

'''Definition [Regularity with respect to a partition]'''

We say that a <math> k </math>-graph <math> \mathcal{H}^{(k)} </math> is <math> (\delta_k, r) </math>-regular with respect to a family of partitions <math> \mathcal{P} </math> if all but at most <math> \delta_k n^k </math> <math> k- </math>edges <math> K </math> of <math> \mathcal{H}^{(k)} </math> have the property that <math> K \in \mathcal{K}_k(\mathcal{G}^{(1)}) </math> and if <math> \mathbf{P} = \{P^{(j)}\}_{j=1}^{k-1} </math> is unique <math> (k, k-1) </math>-complex for which <math> K \in \mathcal{K}_k(P^{(k-1)}) </math>, then <math> \mathcal{H}^{(k)} </math> is <math> (\delta_k, d(\mathcal{H}^{(k)} \vert P^{(k-1)}), r) </math> regular with respect to <math> \mathcal{P} </math>.</blockquote>

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