A Ramsey-type result for the hypercube

Alon, Noga; Radoičić, Radoš; Sudakov, Benny; Vondrák, Jan

doi:10.1002/jgt.20181

Cited by 32 publications

(46 citation statements)

References 7 publications

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“…The proof follows ideas of Alon et al [4]. Consider a hypercube Q n , with sufficiently large n > 6m to be determined later, and some arbitrary 2-edgecoloring χ : E(Q n ) → {blue, red}.…”

Section: Ramsey-type Theoremmentioning

confidence: 97%

“…We are inspired by a Ramsey-type result due to Alon et al [4], in which they show that, for any given length ≥ 5, any r-edge coloring of a high dimensional hypercube contains a monochromatic cycle of length 2 . Unfortunately, we cannot immediately use their results, but we show a similar Ramsey-type result for a different, carefully constructed structure; we assert that every 2-edge coloring of high dimensional hypercubes Q n contains a monochromatic copy of that structure.…”

Section: Example 12 (Generalized Weighted Shapley)mentioning

confidence: 99%

“…W.l.o.g. each player is associated with a distinct vertex 4 . For any graph G and any set of players N , a cost-sharing protocol Ξ assigns, for every e ∈ E, some cost-sharing method ξ e on N .…”

Section: Model and Definitionsmentioning

confidence: 99%

“…, s − 1, s}, but when r = 1, we simply write [s]. We follow definitions and notation of [4]. Let Q n be the graph of the n-dimensional hypercube whose vertex set is {0, 1}…”

Section: Proof Overviewmentioning

confidence: 99%

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Designing Networks with Good Equilibria under Uncertainty

Christodoulou¹,

Sgouritsa

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the problem of designing network costsharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree.We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model).In the adversarial model, the goal of the designer is to choose a single, universal cost-sharing protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design?We first demonstrate that there exist classes of graphs where knowledge of the underlying metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than Ω(log k). Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of Ω(log k), which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes.Then we switch to the stochastic model, where each player is independently activated according to some probability distribution that is known to the * Department of Computer Science, University of Liverpool, UK. Email:gchristo@liverpool.ac.uk. This work was supported by EP/M008118/1, EP/K01000X/1 and LT140046.† Department of Computer Science, University of Liverpool, UK. Email:salkmini@liverpool.ac.uk designer. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark, that the first result holds also for the black-box model, where the probabilities are not known to the designer, but is allowed to draw independent (polynomially many) samples.

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Section: Ramsey-type Theoremmentioning

confidence: 97%

Section: Example 12 (Generalized Weighted Shapley)mentioning

confidence: 99%

Section: Model and Definitionsmentioning

confidence: 99%

“…, s − 1, s}, but when r = 1, we simply write [s]. We follow definitions and notation of [4]. Let Q n be the graph of the n-dimensional hypercube whose vertex set is {0, 1}…”

Section: Proof Overviewmentioning

confidence: 99%

See 2 more Smart Citations

Designing Networks with Good Equilibria under Uncertainty

Christodoulou¹,

Sgouritsa

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…On the other hand it is shown in [1] that for any fixed k, in any k-coloring of the edges of a sufficiently large cube there are monochromatic cycles of every even length greater than 6. Note, however, that the Turán problem for cycles of length 4k + 2 is still wide open.…”

Section: Introductionmentioning

confidence: 99%

Turán’s Theorem in the Hypercube

Alon¹,

Krech²,

Szabó³

2007

SIAM J. Discrete Math.

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We are motivated by the analogue of Turán's theorem in the hypercube Q n : how many edges can a Q d -free subgraph of Q n have? We study this question through its Ramsey-type variant and obtain asymptotic results. We show that for every odd d it is possible to color the edges of Q n withcolors, such that each subcube Q d is polychromatic, that is, contains an edge of each color. The number of colors is tight up to a constant factor, as it turns out that a similar coloring with d+1 2 + 1 colors is not possible. The corresponding question for vertices is also considered. It is not possible to color the vertices of Q n with d + 2 colors, such that any Q d is polychromatic, but there is a simple d + 1 coloring with this property. A relationship to anti-Ramsey colorings is also discussed.We discover much less about the Turán-type question which motivated our investigations. Numerous problems and conjectures are raised.

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Inducibility in the hypercube

Goldwasser,

Hansen

2023

Journal of Graph Theory

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Let be the hypercube of dimension and let and be subsets of the vertex set , called configurations in . We say that is an exact copy of if there is an automorphism of which sends onto . Let be an integer, let be a configuration in and let be a configuration in . We let be the maximum, over all configurations in , of the fraction of sub‐‐cubes of in which is an exact copy of , and we define the ‐cube density of to be the limit as goes to infinity of . We determine for several configurations in and as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs.

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A Ramsey-type result for the hypercube

Cited by 32 publications

References 7 publications

Designing Networks with Good Equilibria under Uncertainty

Designing Networks with Good Equilibria under Uncertainty

Turán’s Theorem in the Hypercube

Inducibility in the hypercube

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