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The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Orthogonal Latin Squares Ask Question. Asked 8 years, 6 months ago.

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Maybe I would have additional comments here. The problem came to be while reading some articles on finite geometry. I began wondering if anybody had previously studied this. For example. For example, the following are two orthogonal Latin squares of order 3. I want to understand some properties of this graph. It is known since from Tarry hand checking all Latin squares of order 6 that no two Latin squares of order 6 are mutually orthogonal.

Reposted answer from MathSE; seems like there's a little more attention here. Brendan McKay's comment settles the conjecture, and you addressed the coloring question. Here I have some comments on maximum degree. There's still the cycle question The maximum degree in the graph is likely unbounded. His most striking result concerns the square displayed in Figure The existence or non-existence of such triads remains an open question.

In fact, that particular square has 12, orthogonal mates Maenhaut and Wanless, J.

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**Subhash Khot: Characterization of Non-expanding Sets in the Grassmann Graph**

Asked 6 months ago. Active 5 months ago. Viewed 81 times. I would welcome any intuition, direction for some articles, or any known additional facts.

Thomas Lesgourgues. Thomas Lesgourgues Thomas Lesgourgues 6 6 bronze badges.Rank Bounds on the Independence Number. Rainbow arithmetic progressions and anti-van der Waerden numbers.

Hamiltonicity below Dirac's condition. Non-intersecting Ryser hypergraphs. Intervals in the Hales-Jewett Theorem. The k-colour Ramsey number of odd cycles via non-linear optimisation. Bootstrap percolation in Ore-type graphs. Enumeration in arithmetic setting. Small simplicial complexes with large torsion in homology. On the discrepancy of permutation families. Sample compression schemes. In this work we investigate weak versions of rainbow saturation. In the usual rainbow saturation problem we can choose an arbitrary ordering of the missing edges and add all the missing edges back to the graph in that order and in arbitrary colours completing a rainbow copy each time we add an edge.

Instead of being able to add the edges in any ordering to the graph, in a weaker setting we ask for only one specific ordering in which we need to be able to add them back. In the uncoloured setting the bound for this weakened version is linear, so the answer needs to be between these bounds.

Moreover, we investigate how the problem changes if we are allowed to choose a colour for every edge that we add. What if we can choose the colours and an ordering? We give answers to those questions. This represents joint work with Shagnik Das. In this talk, we present a new proof of Kleitman's diametric theorem via spectral methods, and discuss some extensions and generalizations of Kleitman's Theorem, as well as a few other related extremal problems.

This extends the recent results for graphs of Liebenau and Wormald. It follows that within our scope, the distribution of the degree sequence of a random k-uniform hypergraph can be approximated by a sequence of independent random variables with the appropriate binomial distribution.

This is joint work with Anita Liebenau and Nick Wormald. Builder builds an edge between two vertices and Painter paints it immediately red or blue.

The online Ramsey number is the minimum number of edges Builder needs to guarantee a win regardless of Painter's strategy. We will show new lower and upper bounds for the online Ramsey number.

These connections enable to export results from geometry to machine learning.

Telemetry not connectingOur first main result is based on a geometric construction by Tracy Hall of a partial shelling of the cross polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners.

This refutes several previous works in machine learning from the past 11 years. In particular, it implies that all previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous.

On the positive side we present a new construction of an unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabaled sample compression scheme extends to ample a. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.

Joint work with Ferdinand Ihringer and Valentina Pepe. In this process, we start with initial "infected" set of edges E 0and infect new edges according to a predetermined rule.

Sr 72 top speedThey also conjectured that the maximal running time is o n 2 for every integer r.This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Its focus is on orthomorphisms and complete mappings of finite groups, while also offering a complete proof of the Hallâ€”Paige conjecture.

The use of latin squares in constructions of nets, affine planes, projective planes, and transversal designs also motivates this inquiry. The text begins by introducing fundamental concepts, like the tests for determining whether a latin square is based on a group, as well as orthomorphisms and complete mappings.

From there, it describes the existence problem for complete mappings of groups, building up to the proof of the Hallâ€”Paige conjecture. The third part presents a comprehensive study of orthomorphism graphs of groups, while the last part provides a discussion of Cartesian projective planes, related combinatorial structures, and a list of open problems.

Its presentation draws on a basic understanding of finite group theory, finite field theory, linear algebra, and elementary number theoryâ€”more advanced theories are introduced in the text as needed. Since the mid s, his primary research has been on orthomorphisms and complete mappings of finite groups and their applications. These mappings arise in the study of mutually orthogonal latin squares that are derived from the multiplication tables of finite groups.

As an offshoot of this research, he has also worked on graph representations. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Developments in Mathematics Free Preview. Presents the first unified proof of the Hallâ€”Paige conjecture Discusses the actions of groups on designs derived from latin squares Includes an extensive list of open problems on the construction and structure of orthomorphism graphs suitable for researchers and graduate students see more benefits.

Buy eBook. Buy Hardcover. Buy Softcover. FAQ Policy. About this book This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Show all. Pages Evans, Anthony B. Elementary Abelian Groups. I Pages Evans, Anthony B. Problems Pages Evans, Anthony B. Show next xx. Read this book on SpringerLink. Recommended for you. PAGE 1.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. For example. For example, the following are two orthogonal Latin squares of order 3.

I want to understand some properties of this graph. It is known since from Tarry hand checking all Latin squares of order 6 that no two were mutually orthogonal. The maximum degree in the graph is likely unbounded. His most striking result concerns the square displayed in Figure The existence or non-existence of such triads remains an open question.

In fact, that particular square has 12, orthogonal mates Maenhaut and Wanless, J. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Graph built from orthogonal Latin squares Ask Question. Asked 6 months ago. Active 6 months ago. Viewed times. Thomas Lesgourgues. Thomas Lesgourgues Thomas Lesgourgues 3, 1 1 gold badge 8 8 silver badges 29 29 bronze badges. Active Oldest Votes. Brian Hopkins Brian Hopkins 1, 10 10 silver badges 16 16 bronze badges.

Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Responding to the Lavender Letter and commitments moving forward. Related In combinatoricstwo Latin squares of the same size order are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct.

A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality is not strongly related to others that appear in algebra and analysis.

A pair of orthogonal Latin squares has traditionally been called a Graeco-Latin squarealthough that term is now somewhat dated. The arrangement of the s -coordinates by themselves which may be thought of as Latin characters and of the t -coordinates the Greek characters each forms a Latin square.

A Graeco-Latin square can therefore be decomposed into two orthogonal Latin squares. When a Graeco-Latin square is viewed as a pair of orthogonal Latin squares, each of the Latin squares is said to have an orthogonal mate.

In an arbitrary Latin square, a selection of positions, one in each row and one in each column whose entries are all distinct is called a transversal of that square.

The positions containing this symbol must all be in different rows and columns, and furthermore the other symbol in these positions must all be distinct. Hence, when viewed as a pair of Latin squares, the positions containing one symbol in the first square correspond to a transversal in the second square and vice versa.

A given Latin square of order n possesses an orthogonal mate if and only if it has n disjoint transversals. The Cayley table without borders of any group of odd order forms a Latin square which possesses an orthogonal mate.

Thus Graeco-Latin squares exist for all odd orders as there are groups that exist of these orders. Such Graeco-Latin squares are said to be group based. Euler was able to construct Graeco-Latin squares of orders that are multiples of four, [2] and seemed to be aware of the following result. Although recognized for his original mathematical treatment of the subject, orthogonal Latin squares predate Euler.

In the form of an old puzzle involving playing cards[4] the construction of a 4 x 4 set was published by Jacques Ozanam in This problem has several solutions. A common variant of this problem was to arrange the 16 cards so that, in addition to the row and column constraints, each diagonal contains all four face values and all four suits as well. According to Martin Gardnerwho featured this problem in his November Mathematical Games column[6] the number of distinct solutions was incorrectly stated to be 72 by Rouse Ball.

This mistake persisted for many years until the correct value of was found by Kathleen Ollerenshaw. Each of the solutions has eight reflections and rotations, giving solutions in total.

No permutation will convert the two solutions into each other. A problem similar to the card problem above was circulating in St. Petersburg in the late s and, according to folklore, Catherine the Great asked Euler to solve it, since he was residing at her court at the time. A very curious question, which has exercised for some time the ingenuity of many people, has involved me in the following studies, which seem to open a new field of analysis, in particular the study of combinations.

The question revolves around arranging 36 officers to be drawn from 6 different regiments so that they are ranged in a square so that in each line both horizontal and vertical there are 6 officers of different ranks and different regiments. Euler was unable to solve the problem, but in this work he demonstrated methods for constructing Graeco-Latin squares where n is odd or a multiple of 4. The non-existence of order six squares was confirmed in by Gaston Tarry through a proof by exhaustion.

InR. Bose and S. Shrikhande constructed some counterexamples dubbed the Euler spoilers of order 22 using mathematical insights. A set of Latin squares of the same order such that every pair of squares are orthogonal that is, form a Graeco-Latin square is called a set of mutually orthogonal Latin squares or pairwise orthogonal Latin squares and usually abbreviated as MOLS or MOLS n when the order is made explicit. For example, a set of MOLS 4 is given by: [16].

And a set of MOLS 5 : [17]. While it is possible to represent MOLS in a "compound" matrix form similar to the Graeco-Latin squares, for instance.So watching Mary Jane episodes available via this BET app gives me that opportunity to sort of check out "the Woman in the mirror".

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## Mutually orthogonal Latin squares

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### Orthogonal Latin Squares Based on Groups

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## On mutually orthogonal disjoint copies of graph squares