5 research outputs found
The smallest sets of points not determined by their X-rays
Let be an -point set in with
and . A (discrete) X-ray of
in direction gives the number of points of on each line parallel to
. We define as the minimum number for which
there exist directions (pairwise linearly independent and
spanning ) such that two -point sets in exist
that have the same X-rays in these directions. The bound
has been observed many times in the
literature. In this note we show
for . For the
cases and , , this
represents the first upper bound on that is polynomial
in . As a corollary we derive bounds on the sizes of solutions to both the
classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we
establish lower bounds on that enable us to prove a
strengthened version of R\'enyi's theorem for points in
The number of unit distances is almost linear for most norms
We prove that there exists a norm in the plane under which no n-point set
determines more than O(n log n log log n) unit distances. Actually, most norms
have this property, in the sense that their complement is a meager set in the
metric space of all norms (with the metric given by the Hausdorff distance of
the unit balls)
How many points can be reconstructed from k projections?
Let A be an n-point set in the plane. A discrete X-ray of A in direction u gives the number of points of A on each line parallel to u. We define F(k) as the maximum number n such that there exist k directions u1,...,uk such that every set of at most n points in the plane can be uniquely reconstructed from its discrete X-rays in these directions. A simple “cube ” construction shows F(k) ≤ 2 k−1. We establish the lower bound F(k) ≥ 2 Ω(k / log k) by reducing the problem through linear algebra to a graphtheoretic question, for which we then obtain an almost tight bound. As a part of the proof we establish a result in extremal theory that allows one to conclude that, under certain conditions, a graph has only at most a logarithmic density, which may be of independent interest. We also improve the upper bound to F(k) ≤ O(1.81712 k) (or O(1.79964 k) if we allow A to be a multiset)
Essentially tight bounds for rainbow cycles in proper edge-colourings
An edge-coloured graph is said to be rainbow if no colour appears more than
once. Extremal problems involving rainbow objects have been a focus of much
research over the last decade as they capture the essence of a number of
interesting problems in a variety of areas. A particularly intensively studied
question due to Keevash, Mubayi, Sudakov and Verstra\"ete from 2007 asks for
the maximum possible average degree of a properly edge-coloured graph on
vertices without a rainbow cycle. Improving upon a series of earlier bounds,
Tomon proved an upper bound of for this question. Very
recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the
term in Tomon's bound, showing a bound of . We prove an upper
bound of for this maximum possible average degree when
there is no rainbow cycle. Our result is tight up to the term, and so it
essentially resolves this question. In addition, we observe a connection
between this problem and several questions in additive number theory, allowing
us to extend existing results on these questions for abelian groups to the case
of non-abelian groups