scholarly journals Rainbow Ramsey Problems for the Boolean Lattice

Order ◽  
2021 ◽  
Author(s):  
Fei-Huang Chang ◽  
Dániel Gerbner ◽  
Wei-Tian Li ◽  
Abhishek Methuku ◽  
Dániel T. Nagy ◽  
...  
Keyword(s):  
Boolean Lattice ◽  
Ramsey Problem ◽  
Monochromatic Copy

AbstractWe address the following rainbow Ramsey problem: For posets P, Q what is the smallest number n such that any coloring of the elements of the Boolean lattice Bn either admits a monochromatic copy of P or a rainbow copy of Q. We consider both weak and strong (non-induced and induced) versions of this problem.

scholarly journals SYMMETRIC AND ASYMMETRIC RAMSEY PROPERTIES IN RANDOM HYPERGRAPHS

2017 ◽  
Vol 5 ◽  
Author(s):  
LUCA GUGELMANN ◽  
RAJKO NENADOV ◽  
YURY PERSON ◽  
NEMANJA ŠKORIĆ ◽  
ANGELIKA STEGER ◽  
...  
Keyword(s):  
Random Graph ◽  
Upper Bound ◽  
Natural Extension ◽  
Fixed Number ◽  
The Other ◽  
Ramsey Problem ◽  
Uniform Hypergraphs ◽  
General Bound ◽  
Random Hypergraphs ◽  
Monochromatic Copy

A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the ‘0-statement’), while for $p\geqslant Cn^{-1/m_{2}(F)}$ it is $1-o(1)$ (the ‘1-statement’). Here $m_{2}(F)$ denotes the 2-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n,p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ for the 1-statement, where $m_{k}(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if $k\geqslant 4$: there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

scholarly journals Ramsey-type problems in orientations of graphs ⇤

2018 ◽  
Author(s):  
Bruno Pasqualotto Cavalar
Keyword(s):  
Random Graphs ◽  
Oriented Graph ◽  
Ramsey Number ◽  
Threshold Function ◽  
Ramsey Problem ◽  
Minimum Number ◽  
Ramsey Type ◽  
Monochromatic Copy

The Ramsey number R(H) of a graph H is the minimum number n such that there exists a graph G on n vertices with the property that every two-coloring of its edges contains a monochromatic copy of H. In this work we study a variant of this notion, called the oriented Ramsey problem, for an acyclic oriented graph H~ , in which we require that every orientation G~ of the graph G contains a copy of H~ . We also study the threshold function for this problem in random graphs. Finally, we consider the isometric case, in which we require the copy to be isometric, by which we mean that, for every two vertices x, y 2 V (H~ ) and their respective copies x0, y0 in G~ , the distance between x and y is equal to the distance between x0 and y0.

Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size

2003 ◽  
Vol 24 (2) ◽  
pp. 219-228 ◽  
Author(s):  
Tim Hsu ◽  
Mark J. Logan ◽  
Shahriar Shahriari ◽  
Christopher Towse
Keyword(s):  
Boolean Lattice ◽  
Uniform Size

scholarly journals Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 271-300 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Empty Graph ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove a lower bound ofnβ(F,r)on the typical duration of this game, where β(F,r) is a function that is strictly increasing inrand satisfies limr→∞β(F,r) = 2 − 1/m2(F), wheren2−1/m2(F)is the threshold of the corresponding offline colouring problem.

scholarly journals The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 764
Author(s):  
Yaser Rowshan ◽  
Mostafa Gholami ◽  
Stanford Shateyi
Keyword(s):  
Positive Integer ◽  
Ramsey Number ◽  
Complete Multipartite Graph ◽  
Ramsey Numbers ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Monochromatic Copy

For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

scholarly journals On a Ramsey Problem Involving the 3-Pan Graph

2018 ◽  
Vol 16 (2) ◽  
pp. 437-441 ◽  
Author(s):  
Chula Jayawardene ◽  
Keyword(s):  
Ramsey Problem

Flowers and Steps in the Boolean Lattice of Hexagrams

2004 ◽  
Vol 31 (4) ◽  
pp. 489-504
Author(s):  
Andreas Schöter
Keyword(s):  
Boolean Lattice

A FCA-Based Cognitive Diagnosis Model for CAT

2011 ◽  
pp. 151-170
Author(s):  
Yang Shuqun ◽  
Ding Shuliang
Keyword(s):  
Large Scale ◽  
Matrix Theory ◽  
Estimation Method ◽  
Boolean Lattice ◽  
Cognitive Diagnosis ◽  
Remedial Measure ◽  
Concept Lattices ◽  
Knowledge States ◽  
Attribute Hierarchy ◽  
Developing Area

There is little room for doubt about that cognitive diagnosis has received much attention recently. Computerized adaptive testing (CAT) is adaptive, fair, and efficient, which is suitable to large-scale examination. Traditional cognitive diagnostic test needs quite large number of items, the efficient and tailored CAT could be a remedy for it, so the CAT with cognitive diagnosis (CD-CAT) is prospective. It is more beneficial to the students who live in the developing area without rich source of teaching, and distance education is adopted there. CD is still in its infancy (Leighton at el.2007), and some flaws exist, one of which is that the rows/columns could form a Boolean lattice in Tatsuoka’s Q-matrix theory. Formal Concept Analysis (FCA) is proved to be a useful tool for cognitive science. Based on Rule Space Model (RSM) and the Attribute Hierarchy Method (AHM), FCA is applied into CD-CAT and concept lattices are served as the models of CD. The algorithms of constructing Qr matrice and concept lattices for CAT, and the theory and methods of diagnosing examinees and offering the best remedial measure to examinees are discussed in detail. The technology of item bank construction, item selection strategies in CD-CAT and estimation method are considered to design a systemic CD-CAT, which diagnoses examinees on-line and offers remedial measure for examinees in time. The result of Monte Carlo study shows that examinees’ knowledge states are well diagnosed and the precision in examinees’ abilities estimation is satisfied.

scholarly journals Optimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints

2019 ◽  
Vol 57 (2) ◽  
pp. 783-809 ◽  
Author(s):  
Yu-Jui Huang ◽  
Saeed Khalili
Keyword(s):  
Optimal Consumption ◽  
Ramsey Problem

scholarly journals Ramsey properties of randomly perturbed graphs: cliques and cycles

2020 ◽  
Vol 29 (6) ◽  
pp. 830-867 ◽  
Author(s):  
Shagnik Das ◽  
Andrew Treglown
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Graph Model ◽  
Significant Progress ◽  
Random Choice ◽  
Ramsey Problem ◽  
Random Graph Model ◽  
Dense Graph ◽  
Precise Solution ◽  
Analogous Question

AbstractGiven graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

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