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Ilona Palásti (1924–1991) was a Hungarian mathematician who worked at the
Alfréd Rényi Institute of Mathematics The Alfréd Rényi Institute of Mathematics ( hu, Rényi Alfréd Matematikai Kutatóintézet) is the research institute in mathematics of the Hungarian Academy of Sciences. It was created in 1950 by Alfréd Rényi, who directed it until his death. ...
. She is known for her research in
discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...
, geometric probability, and the theory of random graphs. With Alfréd Rényi and others, she was considered to be one of the members of the Hungarian School of Probability.


Contributions

In connection to the Erdős distinct distances problem, Palásti studied the existence of point sets for which the ith least frequent distance occurs i times. That is, in such points there is one distance that occurs only once, another distance that occurs exactly two times, a third distance that occurs exactly three times, etc. For instance, three points with this structure must form an
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
. Any n evenly-spaced points on a line or circular arc also have the same property, but Paul Erdős asked whether this is possible for points in
general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
(no three on a line, and no four on a circle). Palásti found an eight-point set with this property, and showed that for any number of points between three and eight (inclusive) there is a subset of the hexagonal lattice with this property. Palásti's eight-point example remains the largest known. Another of Palásti's results in discrete geometry concerns the number of triangular faces in an arrangement of lines. When no three lines may cross at a single point, she and
Zoltán Füredi Zoltán Füredi (Budapest, Hungary, 21 May 1954) is a Hungarian mathematician, working in combinatorics, mainly in discrete geometry and extremal combinatorics. He was a student of Gyula O. H. Katona. He is a corresponding member of the Hungarian ...
found sets of n lines, subsets of the diagonals of a regular 2n-gon, having n(n-3)/3 triangles. This remains the best lower bound known for this problem, and differs from the upper bound by only O(n) triangles. In geometric probability, Palásti is known for her conjecture on random sequential adsorption, also known in the one-dimensional case as "the parking problem". In this problem, one places non-overlapping balls within a given region, one at a time with random locations, until no more can be placed. Palásti conjectured that the average packing density in d-dimensional space could be computed as the dth power of the one-dimensional density. Although her conjecture led to subsequent research in the same area, it has been shown to be inconsistent with the actual average packing density in dimensions two through four. Palásti's results in the theory of random graphs include bounds on the probability that a random graph has a Hamiltonian circuit, and on the probability that a random
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pai ...
is
strongly connected In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that ...
.


Selected publications


References

{{DEFAULTSORT:Palasti, Ilona 1924 births 1991 deaths 20th-century Hungarian mathematicians Women mathematicians Graph theorists Probability theorists