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By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.
The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.
Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons
$X_1, X_2, \ldots $
, where the
$X_i$
are independent random variables with distribution given by X. The covering time T is the smallest integer
$t\geq 0$
such that
$\bigcup_{i=1}^t X_i=V$
. The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.
In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible.
This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.
We show that for all
$m,k,r\in \mathbb{N}$
, there is an
$n\in \mathbb{N}$
such that whenever
$L$
is a Latin square of order
$m$
and the Cartesian product
$L^{n}$
of
$n$
copies of
$L$
is
$r$
-coloured, there is a monochrome Latin subsquare of
$L^{n}$
, isotopic to
$L^{k}$
. In particular, for every prime
$p$
and for all
$k,r\in \mathbb{N}$
, there is an
$n\in \mathbb{N}$
such that whenever the multiplication table
$L({\mathbb{Z}_{p}}^{n})$
of the group
${\mathbb{Z}_{p}}^{n}$
is
$r$
-coloured, there is a monochrome Latin subsquare of order
$p^{k}$
. On the other hand, we show that for every group
$G$
of order
$\leq 15$
, there is a 2-colouring of
$L(G)$
without a nontrivial monochrome Latin subsquare.
Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice.
We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph
$K_{16}$
. Of these, 89 have a nontrivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [‘Perfect 1-factorisations of
$K_{16}$
with nontrivial automorphism group’, J. Combin. Math. Combin. Comput.47 (2003), 97–111]). We also (i) describe a new invariant which distinguishes between the P1Fs of
$K_{16}$
, (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.
We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if
$A\subseteq \mathbb{F}_{p}$
satisfies
$|A|\leq p^{64/117}$
then
$\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$
Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018, arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy
$E^{+}(P)$
of some subset
$P\subseteq A+A$
. Our main novelty comes from reducing the estimation of
$E^{+}(P)$
to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.
We construct a family of self-affine tiles in
$\mathbb{R}^{d}$
(
$d\geqslant 2$
) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in
$\mathbb{R}^{2}$
, and its extension to
$\mathbb{R}^{3}$
by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.
An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.
Le diagrams and Grassmann necklaces both index the collection of positroids in the nonnegative Grassmannian Gr≥0(k, n), but they excel at very different tasks: for example, the dimension of a positroid is easily extracted from its Le diagram, while the list of bases of a positroid is far more easily obtained from its Grassmann necklace. Explicit bijections between the two are, therefore, desirable. An algorithm for turning a Le diagram into a Grassmann necklace already exists; in this note, we give the reverse algorithm.
We discuss 1-factorizations of complete graphs that “match” a given Hadamard matrix. We prove the existence of these factorizations for two families of Hadamard matrices: Walsh matrices and certain Paley matrices.
The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider
$m$
runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least
$1/m$
from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever
$n\geqslant 3$
, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the
$n$
-dimensional Euclidean space.
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.
It was shown by Gruslys, Leader and Tan that any finite subset of
$\mathbb{Z}^{n}$
tiles
$\mathbb{Z}^{d}$
for some
$d$
. The first non-trivial case is the punctured interval, which consists of the interval
$\{-k,\ldots ,k\}\subset \mathbb{Z}$
with its middle point removed: they showed that this tiles
$\mathbb{Z}^{d}$
for
$d=2k^{2}$
, and they asked if the dimension needed tends to infinity with
$k$
. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles
$\mathbb{Z}^{4}$
.
We characterize Hermitian cones among the surfaces of degree
$q+1$
of
$\text{PG}(3,q^{2})$
by their intersection numbers with planes. We then use this result and provide a characterization of nonsingular Hermitian varieties of
$\text{PG}(4,q^{2})$
among quasi-Hermitian ones.
We revisit the coordinatisation method for projective planes by considering the consequences of using finite fields to coordinatise projective planes of prime power order. This leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider how the Lenz–Barlotti type of the plane being coordinatised impacts the form of the PTR polynomial, thereby deriving further restrictions.
We prove the existence and give constructions of a
$(p(k)-1)$
-fold perfect resolvable
$(v,k,1)$
-Mendelsohn design for any integers
$v>k\geq 2$
with
$v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$
such that there exists a finite Frobenius group whose kernel
$K$
has order
$v$
and whose complement contains an element
$\unicode[STIX]{x1D719}$
of order
$k$
, where
$p(k)$
is the least prime factor of
$k$
. Such a design admits
$K\rtimes \langle \unicode[STIX]{x1D719}\rangle$
as a group of automorphisms and is perfect when
$k$
is a prime. As an application we prove that for any integer
$v=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}\geq 3$
in prime factorisation and any prime
$k$
dividing
$p_{i}^{e_{i}}-1$
for
$1\leq i\leq t$
, there exists a resolvable perfect
$(v,k,1)$
-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if
$k$
is even and divides
$p_{i}-1$
for
$1\leq i\leq t$
, then there are at least
$\unicode[STIX]{x1D711}(k)^{t}$
resolvable
$(v,k,1)$
-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where
$\unicode[STIX]{x1D711}$
is Euler’s totient function.
This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.