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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.
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.
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.
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.
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.
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.
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.
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 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.
Let
$A$
be a set of natural numbers. Recent work has suggested a strong link between the additive energy of
$A$
(the number of solutions to
$a_{1}+a_{2}=a_{3}+a_{4}$
with
$a_{i}\in A$
) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of
$A$
modulo
$1$
. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.
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.
We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci.126 (2016), 591–612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from θ-covers, and consider related problems.
It is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.
This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk(
$\sqrt{n}$
), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θk(nk/(k+1)).
The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group
$G$
preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on
$G$
, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that
$G$
cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.