Turán-type results for intersection graphs of boxes

Tomon, István; Zakharov, Dmitriy

doi:10.1017/s0963548321000171

Cited by 7 publications

(11 citation statements)

References 10 publications

(14 reference statements)

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“…Our presentation is self contained, and the resulting bound is O k(n + m) log d−1 n , see Lemma 2.3. This improves the results of Basit et al [BCS+21] and Tomon and Zakharov [TZ21] (except for the case K 2,2 in the plane, where it matches the bound of Tomon and Zakharov [TZ21]). Using a standard projection argument this also improves the bound of Basit et al [BCS+21] for polytopes formed by the intersection of s translated halfspaces, from O(nk log s n) to O(nk log s−1 n), see Lemma 2.4.…”

Section: Our Resultssupporting

confidence: 88%

“…They also studied the case where the objects are polytopes formed by the intersection of s translated halfspaces, where they show a bound of O(n log s n). Independently, Tomon and Zakharov [TZ21] showed a bound of O(kn log 2d+3 ) for the case m = n in R d . They also showed a better bound for the special case d = 2, where their bound is O(n log n) if the intersection graph avoids K 2,2 .…”

Section: Introductionmentioning

confidence: 99%

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On the Number of Incidences when Avoiding the Klan

Har-Peled¹

2021

Preprint

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We prove a bound of O(k(n + m) log d−1 ) on the number of incidences between n points and m axis parallel boxes in R d , if no k boxes contain k common points. That is, the incidence graph between the points and the boxes does not contain K k,k as a subgraph. This new bound improves over previous work by a factor of log d n, for d > 2.We also study the variant of the problem for points and halfspaces, where we use shallow cuttings to get a near linear bound in two and three dimensions.

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Section: Our Resultssupporting

confidence: 88%

Section: Introductionmentioning

confidence: 99%

On the Number of Incidences when Avoiding the Klan

Har-Peled¹

2021

Preprint

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“…Remark 1.4. A bound similar to Theorem (B.1) and an improved bound for Theorem (A.1) in the 2,2free case are established independently by Tomon and Zakharov in [22], in which they also use our Theorem (A.3) to provide a counterexample to a conjecture of Alon et al [1] about the number of edges in a graph of bounded separation dimension, as well as to a conjecture of Kostochka from [13]. Some further Ramsey properties of semilinear graphs are demonstrated by Tomon in [21].…”

mentioning

confidence: 72%

Zarankiewicz’s problem for semilinear hypergraphs

Basit

Chernikov

Starchenko

et al. 2021

Forum of Mathematics, Sigma

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A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert = n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation E consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $d_1 + d_2$ variables for some s. We show that for a fixed k, the number of edges in a $K_{k,k}$ -free semilinear H is almost linear in n, namely $\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$ for any $\varepsilon> 0$ ; and more generally, $\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$ for a $K_{k, \dotsc ,k}$ -free semilinear r-partite r-uniform hypergraph. As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis-parallel sides in $\mathbb {R}^d$ such that their incidence graph is $K_{k,k}$ -free, there can be at most $O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o-minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

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“…It is known that point-box incidence graphs are box intersection graphs in R 3 (see, e.g., [9]). Summarising, we derive the main result of this section.…”

Section: Functionality Of Box Intersection Graphs In R 3 Is Unboundedmentioning

confidence: 99%

Functionality of box intersection graphs

Dallard¹,

Milanič²,

Štorgel³

et al. 2023

Preprint

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Functionality is a graph complexity measure that extends a variety of parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In the present paper, we show that functionality is bounded for box intersection graphs in R 1 , i.e. for interval graphs, and unbounded for box intersection graphs in R 3 . We also study a parameter known as symmetric difference, which is intermediate between twin-width and functionality, and show that this parameter is unbounded both for interval graphs and for unit box intersection graphs in R 2 .

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Turán-type results for intersection graphs of boxes

Cited by 7 publications

References 10 publications

On the Number of Incidences when Avoiding the Klan

On the Number of Incidences when Avoiding the Klan

Zarankiewicz’s problem for semilinear hypergraphs

Functionality of box intersection graphs

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