On the boundary of the region defined by homomorphism densities

Hatami, Hamed; Norin, Sergey

doi:10.4310/joc.2019.v10.n2.a1

Cited by 4 publications

(3 citation statements)

References 7 publications

(11 reference statements)

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“…Regarding our results on the shape of Ω(F), there are (at least) two previous works of a similar flavor: Razborov [22] determined the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs (and showed that the boundary is almost everywhere differentiable) and Hatami-Norine [8] constructed examples which show that the restrictions of the boundary to certain hyperplanes of the region defined by the homomorphism densities of a list of given graphs can have nowhere differentiable parts.…”

Section: Introductionmentioning

confidence: 58%

The feasible region of hypergraphs

Liu

Mubayi

2019

Preprint

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Let F be a family of r-uniform hypergraphs. The feasible region Ω(F ) of F is the set of points (x, y) in the unit square such that there exists a sequence of F -free r-uniform hypergraphs whose edge density approaches x and whose shadow density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x, y) ∈ Ω(F ) is the Turán density π(F ), and Ω(∅) gives the Kruskal-Katona theorem.We undertake a systematic study of Ω(F ), and prove that Ω(F ) is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an F for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of Fisher and Ryan to hypergraphs and extend a classical result of Bollobás by almost completely determining the feasible region for cancellative triple systems.

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Section: Introductionmentioning

confidence: 58%

The feasible region of hypergraphs

Liu

Mubayi

2019

Preprint

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“…For larger graphs the problem becomes extremely challenging. Some general results on the hardness of determining T (F) were obtained by Hatami and Norine in [16,17].…”

Section: Discussionmentioning

confidence: 87%

The Exact Minimum Number of Triangles in Graphs With Given Order and Size

Liu

Pikhurko

Staden

2020

Forum of Mathematics, Pi

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What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, the first non-trivial case of this problem was solved by Rademacher in 1941, and the problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from 1, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs. 4.2.

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“…In contrast to Theorem 1.2 Hatami and Norin [ 9 ] gave an example of a finite family F of graphs such that the intersection of the graph profile T pFq with some hyperplane has a nowhere differentiable boundary.…”

Section: General Resultsmentioning

confidence: 99%

The feasible region of induced graphs

Liu,

Mubayi,

Reiher

2021

Preprint

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The feasible region Ω ind pF q of a graph F is the collection of points px, yq in the unit square such that there exists a sequence of graphs whose edge densities approach x and whose induced F -densities approach y. A complete description of Ω ind pF q is not known for any F with at least four vertices that is not a clique or an independent set.The feasible region provides a lot of combinatorial information about F . For example, the supremum of y over all px, yq P Ω ind pF q is the inducibility of F and Ω ind pK r q yields the Kruskal-Katona and clique density theorems.We begin a systematic study of Ω ind pF q by proving some general statements about the shape of Ω ind pF q and giving results for some specific graphs F . Many of our theorems apply to the more general setting of quantum graphs. For example, we prove a bound for quantum graphs that generalizes an old result of Bollobás for the number of cliques in a graph with given edge density. We also consider the problems of determining Ω ind pF q when F " K ŕ , F is a star, or F is a complete bipartite graph. In the case of K ŕ our results sharpen those predicted by the edge-statistics conjecture of Alon et. al. while also extending a theorem of Hirst for K 4 that was proved using computer aided techniques and flag algebras. The case of the 4-cycle seems particularly interesting and we conjecture that Ω ind pC 4 q is determined by the solution to the triangle density problem, which has been solved by Razborov.

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On the boundary of the region defined by homomorphism densities

Cited by 4 publications

References 7 publications

The feasible region of hypergraphs

The feasible region of hypergraphs

The Exact Minimum Number of Triangles in Graphs With Given Order and Size

The feasible region of induced graphs

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