Jump numbers, hyperrectangles and Carlitz compositions

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998.A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg 1998Let A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts)

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Constructions for mutually orthogonal frequency hyperrectangles with a prescribed type

AbstractIn this paper, we give two different ways to construct mutually orthogonal frequency hyperrectangles (MOFHR). Firstly, we exhibit sets of linear polynomials over finite fields that represent complete sets of MOFHR of prime power order, which generalize Mullen's method in (G.L. Mullen, Discrete Math. 69 (1988) 79–84). Secondly, a recursive algorithm is given to construct (d+1)-dimensional MOFHR of type t+1 from d-dimensional MOFHR of type t, which generalizes a recursive procedure described in (Laywine et al., Monatsch Math. 119 (1995) 223–238)

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version