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2019 | OriginalPaper | Chapter

Embedding Graphs into Larger Graphs: Results, Methods, and Problems

Authors : Miklós Simonovits, Endre Szemerédi

Published in: Building Bridges II

Publisher: Springer Berlin Heidelberg

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Abstract

Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which are new breakthrough results, or which—for some other reasons—are very close to us. Some results discussed here got stronger emphasis, since they are connected to Lovász (and sometimes to us).

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previous chapter Finding k Partially Disjoint Paths in a Directed Planar Graph

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We shall indicate the given names mostly in case of ambiguity, in cases where there are two mathematicians with the same family name, (often, but not always, father and son). We shall ignore this “convention” for Erdős, Lovász and Turán.

Sometimes we list papers in their time-order, in some other cases in alphabetical order.

The proof of the approximate Loebl–Komlós–Sós conjecture was first attacked in [12] and then proved in a sequence of papers, from Yi Zhao [824], Cooley [202], ... Hladký, Komlós, Piguet, Simonovits, Stein, and Szemerédi [466‐469].

The first result on the Loebl Conjecture was an “approximate” solution of Ajtai, Komlós, and Szemerédi [12].

Essential parts of this survey are connected to Regularity Lemmas, Blow-up lemmas, applications of Absorbing techniques, where again, there are several very important and nice surveys, covering those parts, e.g., Alon [22], Gerke and Steger [392], Komlós and Simonovits [546], Kühn and Osthus [563, 565], Rödl and Ruciński [689], Steger [782], and many others.

For a concise “description” of this topic, see Rödl, Nagle, Skokan, Schacht, and Kohayakawa [687], and also Solymosi [770].

Often called stability number.

The last paragraph of Turán’s original paper is as follows: “...Further on, I learned from the kind communication of Mr. József Krausz that the value of \(d_k(n)\) given on p 438 for \(k=3\) was found already in 1907 by W. Mantel (Wiskundige Opgaven, vol 10, pp. 60–61). I know this paper only from the reference Fortschritte d. Math. vol 38, p. 270.”

Turán’s papers originally written in Hungarian were translated into English after his death, thus [811] contains the English translation of [808].

E.g., we can put \(a_i\) into \({\mathcal A}_i\) if the smallest prime divisor of \(a_i\) is in \((2^t,2^{t+1}]\) and use a slight generalization of Theorem 2.4 to K(m, n).

A longer annotated bibliography of O’Bryant can be downloaded from the Electronic Journal of Combinatorics [160] on Sidon sets.

Mostly we call this result the Kővári-T. Sós-Turán theorem. Here we added the name of Erdős, since [554] starts with a footnote according to which “As we learned after giving the manuscript to the Redaction, from a letter of P. Erdős, he has found most of the results of this paper.” Erdős himself quoted this result as Kővári-T. Sós-Turán theorem.

For Hungarian authors we shall mostly use the Hungarian spelling of their names, though occasionally this may differ from the way their name was printed in the actual publications.

Here “sharp” means that not only the exponent \(2-{1\over p}\) but the value of \(c_{a,b}\) is also sharp.

The sharpness of the multiplicative constant followed from a later result of Füredi.

Later this question was generalized to excluding an arbitrary family of subgraphs, however, that was only a small extension.

The tetrahedron is \(K_4\), covered by Turán theorem.

Lasso is a graph where we attach a path to a cycle. Perhaps nobody considered the lasso-problem carefully, however, very recently Sidorenko solved a very similar problem of the keyrings [750].

If the automorphism group of G is edge-transitive, then either all the edges are critical, or none of them. By the way, in [762], Simonovits discusses these questions in more details, among others, the extremal problems of generalized Petersen graphs.

The definition applies to hypergraphs as well, the triples of the Fano hypergraph are also critical.

For \(p=2\) this was also known (at least, implicitly) by Erdős.

Here (c)\(\rightarrow \)(b) is trivial, and one can prove that (b) implies (c) with \(n_1=n_0+3p\).

Bollobás [117] also contains similar, strongly related results.

There are several earlier results on similar questions, e.g., Yap [819, 820], Diananda and Yap [240], yet they are slightly different, or several papers of A. Street, see [785].

One has to assume that the edge-multiplicity is bounded, otherwise even for the excluded \(K{}_3\) in the Universe of multigraphs we would get arbitrary many edges. As an exception, in the Füredi-Kündgen theorem [378] no such bound is assumed.

And many others, see e.g., Rödl and Rucinski [689], or the much earlier Bermond, Germa, Heydemann, and Sotteau [104], and the corresponding Sects. 8 and 9.

Vera Sós did not call these areas Universes.

Actually, here they formulated this for \(r_3(n)\).

Speaking of arithmetic progressions we always assume that its terms are distinct.

The conjectures on \(r_k(n)\) were not always correct. Vera Sós wrote a paper [776] on the letters between Erdős and Turán during the war, where one can read that Szekeres e.g., conjectured that for \(n=\frac{1}{2}(3^\ell +1)\) \(r_k(n)\le 2^\ell \). This was later disproved by Behrend [96]. (This conjecture is also mentioned in [323].)

See also Austin [51], Beigleböck [97], Bergelson and Leibman [103] Gowers, [406], Polymath [655],...

Similarly to the proof of \(r_3(n)=o(n)\) from the Ruzsa–Szemerédi Triangle Removal Lemma, (see Theorem 5.26) Rödl, Schacht, Tengen and Tokushige proved \(r_k(n)=o(n)\) and several of its generalizations in [700] “elementarily”, i.e. not using ergodic theoretical tools. On the other hand, they remarked that those days no elementary proof was known on the Density Hales–Jewett theorem.

Using \(\log \log n\) we always assume that \(n\ge 100\), and therefore \(\log \log n>3/2\).

Subsubsections will also be called Subsections.

The description of the typical structure is a stronger result than just counting them.

If (1) is violated then \(a_ia_ja_ka_\ell \) is a square.

\(\mathbf{ex}(n,m,{\mathcal L})\) is the maximum number of edges an \({\mathcal L}\)-free graph \(G\subset K(n,m)\) can have. This problem may produce surprising phenomena when \(n=o(m)\).

The Margulis–Lubotzky–Phillips–Sarnak papers are eigenvalue-extremal, however, as Alon pointed out, (see the last pages of [599]), these constructions are “extremal” for many other graph problems as well.

The Lovász Local Lemma is one of the most important tools in Probabilistic Combinatorics (including the application of probabilistic methods). Its proof is very short, and it is described, among others, in the Alon–Spencer book [48], in Spencer [778], or in the original paper, available at the “Erdős homepage” [827].

One problem with this sentence is that the notion of “construction” is not well defined, one of us witnessed a discussion between Erdős and another excellent mathematician about this, but they strongly disagreed. As to the constructions, we mention the Frankl–Wilson construction of Ramsey graphs [359], or some papers of Barak, Rao, Shaltiel, and Wigderson [89] and others.

A generalization of these graphs is the generalized random graph, where we join the two vertices with probability \(p_{ij}\), independently.

The Ramsey numbers R(L, M) form a twice infinite matrix whose rows and columns are indexed by the graphs L and M. If \(L\ne M\), then R(L, M) is called “off-diagonal”.

We mention just a few related papers, for a more detailed description of this area see the remarks of Simonovits in [811], and the surveys of Katona [490, 491].

This is equivalent to that deleting any 5 vertices of \(D_{12}\) one gets \(\ge 3\)-chromatic graphs.

The Master Thesis of Warnke contained results on \(K_4\).

Again, there is some difference between the cases of even and odd n.

Erdős’ paper contains many further interesting and important results.

This theorem may remind us of the Removal Lemma, (see Sect. 5.4) yet, it is different in several aspects. Both they assert that either we have many copies of L in \(G{}_n\), or we can get an L-free graph from \(G{}_n\) by deleting a few edges. However, the Removal Lemma has no condition on \(e(G{}_n)\) and the Lovász–Simonovits theorem provides a much stricter structure.

This result can also be used for negative values of k, (and sometimes we need this), however, then we should replace k by |k| in some of the formulas.

Actually, the First Moment Method yields a little better estimate, but far from being satisfactory. One can also see that as a is fixed and b gets larger, the “random construction” exponents converge to the optimal one. This motivates, among others, [42, 532].

This choice ensured that the neighbourhoods did not contain three collinear points.

The original version claimed a slightly better estimate.

The union of two complete graphs of n/2 vertices having at most one common vertex in common also show the sharpness. For \(=2\ell -1\) one can use \(K(\ell ,\ell -1)\) for sharpness.

See the introduction of [513]. They also point out the applications of this theorem, e.g., in [36, 478] for “deviation bounds” for sums of weekly dependent random variables, and in the Rödl-Ruciński proof of the Blow-up Lemma [688].

In the Gyárfás Conjecture we try to pack many different trees into a complete graph.

Or, in other cases edge-disjoint copies. Here “vertex-independent” and “vertex-disjoint” are the same.

Actually, they formulated two “similar” conjectures, we consider only one of them.

We must repeat that the meaning of “to construct them” is not quite well defined. Let us agree for now that the primary aim was to eliminate the randomness.

More precisely, edge-colour-critical graph.

An r-cone is a 3-uniform hypergraph obtained from a cycle \(x_1,\dots ,x_k\) by adding r new vertices \(y_1,\dots ,y_r\) and all the triples \(y_jx_ix_{i+1}\) (where \(x_{k+1}=x_1\)).

In graph-theoretical language, Lovász excluded all the 3-graphs for which the Sperner Lemma holds: for which each pair was contained by an even number of triples.

Observe that this is motivated by [154], and that we formulated it in its simplest case, however, we (more precisely, Nati Linial) meant a whole family of problems. He spoke about them in his talk in the Lovász Conference, 2018.

The same applies to the book of Molloy and Reed [615].

This is a PNAS “survey”, with an accompanying paper of Solymosi [770].

Again, this case differs from the others: if we try to optimize some parameter on all n-vertex graphs, or on the subgraphs if the d-dimensional cube,..., that problem is well defined for individual graphs, while the assertions on the subgraphs of a random graphs make sense only in some asymptotic sense, the assertions always contain the expression “almost surely as \(n\rightarrow \infty \)”.

Often called a Berge k-cycle: in Fig. 14(b) the edges of a \(C_6\) are covered by 3-tuples.

Actually, in [536] one needs to exclude only cycles of length 2, 3, and 4, where a cycle of length 2 is a pair of hyperedges intersecting in at least two vertices. Even this is improved in the next theorem.

Of course, Shearer’s improvement yields an improvement of c in (14).

Actually, the proof works with \(\tilde{c}={1\over 162}-o(1) \).

If three of them are collinear that provides 0.

Here \(f\ll g\) is the same as \(f\le cg\), for some absolute constant \(c>0\).

Actually, Hajnal and Szemerédi found this problem on Gowers’ homepage, but, as it turned out, from [643], originally the problem was asked by Paul Erdős, [285].

Actually, above we spoke about the “diagonal case” but [643] covers some off-diagonal cases too.

In citations we use our numbers, not the original ones.

Perhaps the expression “linear hypergraph” was unknown those days.

The question was that if \(e(G{}_n)=e(T_{n,p})+{\varepsilon }n^2\), how large \({K_{p+1}}(m,\dots ,m)\) can be guaranteed in \(G{}_n\)? This maximum \(m=m(p,{\varepsilon })\) had a very weak estimate in [321]. This was improved to \(c(p,{\varepsilon })\log n\) by Bollobás and Erdős [124], which was improved by Bollobás, Erdős, and Simonovits [127]. Chvátal and Szemerédi needed the Regularity Lemma to get the “final” result, sharp up to a multiplicative absolute constant.

This was formulated by many researchers.

In random graphs this holds for sufficiently large disjoint vertex sets.

Perhaps the name “Reduced Graph” comes from Simonovits, the “Cluster Graph” from Komlós, and the theorem itself was originally called “Uniformity Lemma”: the name “Regularity Lemma” became popular only later.

Estimate \(e(H_\nu )\) or prove some structural property of \(H_\nu \).

Watch out, mostly it does not matter, but here, in case of sharp Ramsey results one has to distinguish the lower and upper Ramsey numbers. The upper one is the smallest one for which there is no good colouring, here \(4k-3\). The lower Ramsey number is \(R(L_1,\dots ,L_r)-1\).

Here by linear we mean O(n).

Actually, Turán’s corresponding results, or the Erdős–Sós-type Ramsey–Turán theorems were not used in “applications”, however the Ajtai–Komlós–Szemerédi-type results are also in this category and, as we have seen in Sects. 4.2–4.10, they were used in several beautiful and important results.

One has to be cautious with this notation, when we write o(n) instead of a function \(f{}(n)\).

These are the graphs we considered in connection with R(n, 3) in Sect. 2.16. There are many such graphs obtained by various, more involved constructions.

Actually, Rödl proved a slightly stronger theorem, answering a question of Erdős, but the original one, Problem 5.22, is still open.

Here we assume that the mindegree is at least 3.

Some applications of the Ruzsa–Szemerédi theorem are given in Sect. 5.5.

With the exception of the next theorem.

Actually, this assertion is somewhat more involved, see the introduction of [45].

Watch out, some of these papers, e.g., [328] are from before Füredi’s result, some others are from after it.

The corresponding extremal value will be denoted by \({\chi _S}(n,E,L)\). Here S stands for “strong” in \({\chi _S}\). It is the strong chromatic number of the v(L)-uniform hypergraph whose hyperedges are the v(L)-sets of vertices of the copies of \(L\subset G{}_n\).

Here “almost surely” means that its probability tends to 1 as \(n\rightarrow \infty \).

See Meta-Theorem 2.19.

Rödl also knew it, but it seems that he had not published it.

The “edit” distance is the same used in [751]: the minimum number of edges to be changed to get from \(G{}_n\) a graph isomorphic to \(H_n\).

Though we formulate a theorem on the local resilience of graphs for some graph property, we shall not define here the notion of local and global resilience: we refer the reader to the papers of Sudakov and Vu [787], or Balogh, Csaba, and Samotij [68], or suggest to read only Theorem 5.47.

We have defined only the quasi-random graphs here, for pseudo-random graphs see e.g., [559, 803], for expanders see e.g., [20].

Here we have \(k+1\) classes, since originally there was also an exceptional class \(V_0\), different from the others. This \(V_0\) can be forgotten: its vertices can be distributed in the other classes.

The theorem also has a version on parallel computation.

100

Here we do not define the “steps” and ignore again the difference caused by neglecting \(V_0\) in Theorem 5.3.

101

As we have mentioned, this is not quite true. It was invented to prove a conjecture of Bollobás, Erdős, and Simonovits on the parametrized Erdős–Stone theorem, and was first used in the paper of Chvátal and Szemerédi [188]. A weaker, bipartite, asymmetric version of it was used to prove that \(r_k(n)=o(n)\).

102

Principal component analysis, see e.g., Frieze, Kannan, Vempala, and Drineas [246, 367].

103

There are many results showing that the number of clusters must be very large. The first such result is due to Gowers [401].

104

They call it d-arrangeable.

105

One form of this is expressed in the Combinatorial Nullstellensatz of Alon [23].

106

Two remarks should be made here: (a) Originally Property Testing was somewhat different, see e.g., Goldreich, Goldwasser, and Ron [399]. (b) The theory of graph limits also has a part investigating property testing, see e.g., [138, 140],..., [596].

107

Actually, J. Fox eliminated the application of the Regularity Lemma from the proof of the Triangle Removal Lemma, see Sect. 5.4 or [347].

108

Actually, this was the motivation for Pósa. (Later the Hajnal-Szemerédi theorem was proved in simpler ways.)

109

In some cases we have only weaker conclusions, e.g., in the Bondy–Simonovits theorem [136], and also it may happen in some cases that we get only even cycles! See also the paper of Brandt, Faudree, and Goddard [147] on weakly pancyclic graphs.

110

Kierstead, Kostochka and others have several results where Ore-type conditions imply Hajnal–Szemerédi-type theorem [509, 512], or a Brooks-type theorem [511].

111

The circumference is the length of the longest cycle. Here we exclude the trees.

112

A stronger statement is Theorem 7.19.

113

In several cases we must distinguish subcases also by some divisibility conditions: not only the proofs but the results also strongly depend on some divisibility conditions.

114

The paper has an Appendix written by Reiher and Schacht, about a version of this problem, also using the Absorption technique. In this version they replace the condition that any linear sized vertex-set contains an edge by a condition that any linear sized set contains “many edges”.

115

Here we took \({\mathcal L}_i={\mathcal L}\).

116

In some sense, this is used also in the original proof of Erdős–Stone theorem [321].

117

For some related result for random or quasi-random graphs see [395, 398, 547, 558], and many others.

118

Here “\(G[V_i]\) is Hamiltonian” means that it has a spanning cycle.

119

We used vertex-colouring in connection with colouring properties of excluded subgraphs, or equipartitions in Hajnal–Szemerédi theorem, ...

120

In fact, one can allow the first o(n) trees to have arbitrary degrees.

121

I.e. \(T_1,\dots ,T_n\) pack into \(K{}_N\).

122

We used superscript since here \(v(T^m)\) is not necessarily m.

123

More generally, we may fix for each colour i a family \({\mathcal L}_i\) and may try to cover \(V(G{}_n)\) by vertex disjoint subgraphs \(H_i\in {\mathcal L}_i\).

124

Formally we have here two problems, one when we r-colour \(E(K{}_N)\), the other when we r-colour \(E(G{}_n)\), however, the difference “disappears” if r is large. Further, we may also ask for the largest subgraph \(H\subseteq K{}_N\) that is coloured by at most t colours, which is different from asking for the largest number of edges covered by t monochromatic \(H_i\in {\mathcal L}\).

125

If for a fixed x, all edges xy are Red, and the other edges are Blue, then we need this.

126

Here we use L for an excluded graph, \({\mathbb L}\) for a hypergraph, and \({\mathcal L}\) for a family of graphs or hypergraphs.

127

The strong chromatic number \(\chi _S({\mathbb F_{}^{(r)}})\) of \({\mathbb F_{}^{(r)}}\) is the minimum \(\ell \) for which the vertices of \({\mathbb F_{}^{(r)}}\) can be \(\ell \)-coloured so that each hyperedge gets r distinct colours. Our condition is equivalent with that some \({\mathbb F_{}^{(r)}}\in {\mathcal L}^{(r)}\) is a subgraph of \({\mathbb K_{r}^{(r)}}(a,\dots ,a)\) for some large a.

128

These constructions seem to be forgotten, “lost” and are not that important.

129

Since for hypergraphs we have at least two popular chromatic numbers, therefore the expression r-uniform \(\ell \)-partite may have at least two meanings in the related literature.

130

Generally, \({\mathcal L}_{k,\ell }^{(r)}\) is the family of r-uniform hypergraphs of k vertices and \(\ell \) hyperedges. As we have mentioned, the problem of \(\mathbf{ex}(n,{\mathcal L}_{k,\ell }^{(r)})\) was considered in two papers of Brown, Erdős, and Sós [154, 155] and turned out to be very important in this field. Originally Erdős conjectured a relatively simple asymptotic extremal structure, for \({\mathcal L}_{4,3}^{(3)}\) but his conjecture was devastated by a better construction of Frankl and Füredi [355]. This construction made this problem rather hopeless.

131

Defined after Definition 9.2.

132

Hamiltonicity for Berge cycles were discussed by Bermond, Germa, Heydemann, and Sotteau [104]. Perhaps the first Tight Hamiltonicity was discussed in [492], however, the tight Hamilton cycles were called there Hamilton chains.

133

The breakthrough result of Łuczak [600] was improved by Kohayakawa, Simonovits, and Skokan [530] and generalized by Jenssen and Skokan [479], see also Gyárfás, Ruszinkó, Sárközy, and Szemerédi [432] and Benevides and Skokan [99] and others.

134

For Berge cycle Hamiltonicity there were earlier results, e.g., by Bermond, Germa, Heydemann, and Sotteau [104].

135

They called it chain.

136

Identical with Theorems A,B, above.

137

This, with a sketch of the proof, is nicely explained by Rödl and Ruciński, in [689]. The introduction of [674] and a survey of Yi Zhao [825] are also good descriptions of the otherwise fairly complicated situation.

138

This implies Theorem 9.17.

139

The earlier excellent surveys of Füredi [369, 372] are primarily on small excluded subgraphs.

140

See also Keevash, [502], Barber, Kühn, Lo, and Osthus [93], Barber, Kühn, Lo, Montgomery, and Osthus [92],...

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Miklós Simonovits’ homepage is: www.renyi.hu/~miki

Title: Embedding Graphs into Larger Graphs: Results, Methods, and Problems
Authors: Miklós Simonovits
Endre Szemerédi
Publisher: Springer Berlin Heidelberg
Book: Building Bridges II
Print ISBN: 978-3-662-59203-8

Electronic ISBN: 978-3-662-59204-5

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-662-59204-5_14

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