Saturation problems in the Ramsey theory of graphs, posets and point sets

Damásdi, Gábor; Keszegh, Balázs; Malec, David; Tompkins, Casey; Wang, Zhiyu; Zamora, Oscar

doi:10.1016/j.ejc.2021.103321

“…For linked matchings with at most three links and at most one minedge inside each link edge (for exact definitions see later) we determine their saturation function. Most of these small cases have very specific proofs 7 . Generalizing one of these cases we get an infinite family of ordered graphs with bounded saturation function.…”

Section: Our Resultsmentioning

confidence: 99%

“…Besides graphs there are many further settings where in addition to the extremal problems the saturation problems are also investigated, such as set systems and 0-1 matrices among others [7,8,10,11,15,19]. The case of 0-1 matrices leads us to the case of ordered graphs, our aim is to investigate their saturation functions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Saturation of Ordered Graphs

Bošković

¹

,

Keszegh

²

2023

SIAM J. Discrete Math.

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Recently, the saturation problem of 0-1 matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs.We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of Pálvölgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function.In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.

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“…Another potential direction for future research is saturation functions. Saturation problems have been studied for decades, with much of the focus on graphs [6,8,10,34], posets [12,31], and set systems [11,28]. Recently a saturation function was introduced for forbidden 0-1 matrices [2].…”

Section: Saturation Functions For Subsets Of Rmentioning

confidence: 99%

Continuous Turán numbers

Geneson

2021

Preprint

0

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One of the most famous open problems in combinatorics is the Zarankiewicz problem, which asks for the maximum number of ones in an n × n matrix that has no s × t submatrix of all ones. The Kővari-Sós-Turán theorem provides an upper bound of O(s) for this problem for fixed t ≥ 2, which is known to be sharp in some cases. The Zarankiewicz problem is a subproblem of the more general problem of determining the maximum number of ones ex(n, M ) in an n × n 0-1 matrix that avoids the forbidden 0-1 matrix M .In this paper, we define a notion of containment and avoidance for subsets of R 2 . Then we introduce a new, continuous and super-additive extremal function for subsets P ⊆ R 2 called opx(n, P ), which is the supremum of µ 2 (S) over all open P -free subsets S ⊆ [0, n] 2 , where µ 2 (S) denotes the Lebesgue measure of S in R 2 . We show that opx(n, P ) fully encompasses ex(n, M ) up to a constant factor. More specifically, we define a natural correspondence between finite subsets P ⊆ R 2 and 0-1 matrices M P , and we prove that opx(n, P ) = Θ(ex(n, M P )) for all finite subsets P ⊆ R 2 , where the constants in the bounds depend only on the distances between the points in P .We also discuss bounded infinite subsets P for which opx(n, P ) grows faster than ex(n, M ) for all fixed 0-1 matrices M . In particular, we show that opx(n, P ) = Θ(n 2 ) for any open subset P ⊆ R 2 . We prove an even stronger result, that if Q P is the set of points with rational coordinates in any open subset P ⊆ R 2 , then opx(n, Q P ) = Θ(n 2 ). Finally, we obtain a strengthening of the Kővari-Sós-Turán theorem that applies to infinite subsets of R 2 . Specifically, for subsets P s,t,c ⊆ R 2 consisting of t horizontal line segments of length s with left endpoints on the same vertical line with consecutive segments a distance of c apart, we prove that opx(n, P s,t,c ) = O(s, where the constant in the bound depends on t and c. When t = 2, we show that this bound is sharp up to a constant factor that depends on c. We also extend opx(n, P ) to any number of dimensions, and we generalize most of our results including the strengthening of the Kővari-Sós-Turán theorem to any number of dimensions.

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“…Besides graphs there are many further settings where in addition to the extremal problems the saturation problems are also investigated, such as set systems and 0-1 matrices among others [7,8,10,11,15,19]. The case of 0-1 matrices leads us to the case of ordered graphs, our aim is to investigate their saturation functions.…”

Section: Introductionmentioning

confidence: 99%

Saturation of Ordered Graphs

Bošković¹,

Keszegh²

2022

Preprint

Self Cite

0

View full text Add to dashboard Cite

Recently, the saturation problem of 0-1 matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs.We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of Pálvölgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function.In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.

show abstract

Saturation problems in the Ramsey theory of graphs, posets and point sets

Cited by 6 publications

References 21 publications

Saturation of Ordered Graphs

Saturation of Ordered Graphs

Continuous Turán numbers

Saturation of Ordered Graphs

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