Related papers: Largest set of unitals in projective planes of ord…
In this study we performed a computer search for unitals in planes of order 16. Some new unitals were found and we show that some unitals can be embedded in two or more different planes.
In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one…
The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique…
In this paper, we prove that the largest pure partial plane of order 6 has size 25. At the same time, we classify all pure partial planes of order 6 and size 25 up to isomorphism. Our major approach is computer search. The search space is…
A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a…
A set P of points in R^2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n>=15. Conversely,…
For $v\equiv 1$ or 3 (mod 6), maximum partial triple systems on $v$ points are Steiner triple systems, STS($v$)s. The 80 non-isomorphic STS(15)s were first enumerated around 100 years ago, but the next case for Steiner triple systems was…
We classify the parabolic unitals in regular nearfield planes of odd order $q^2$ whose linear collineation group has the maximal size of $q^3-q$. We also establish a number of more general results concerning parabolic unitals in regular…
We prove that every set of n points in the plane has at most $(16+5/6)^n$ rectangulations. This improves upon a long-standing bound of Ackerman. Our proof is based on the cross-graph charging-scheme technique.
This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the…
In this paper, we study the problem of finding the largest possible set of s points and s lines in a projective plane of order q, such that that none of the s points lie on any of the s lines. We prove that s <= 1+(q+1)(\sqrt{q}-1). We also…
In this paper, we address computational questions surrounding the enumeration of non-isomorphic Andr\'e planes for any prime power order. We are particularly focused on providing a complete enumeration of all such planes for relatively…
A list of different types of a projective line over non-commutative rings with unity of order up to thirty-one inclusive is given. Eight different types of such a line are found. With a single exception, the basic characteristics of the…
We address the following question: what are the cardinalities of maximal finite families of pairwise incident planes in a complex projective space? One proves easily that the span of the planes has dimension 5 or 6. Up to projectivities…
The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a…
There are 123,650 partial groups of order at most 9 and 178,937,003 partial groups of order 10. We explain a computer enumeration of these results and provide a complete list of indecomposable partial groups of order at most 5. We also…
How many squares are spanned by $n$ points in the plane? Here we study the corresponding maximum possible number $S_{\square}(n)$ of squares and determine the exact values for all $n\le 17$. For $18\le n\le 100$ we give lower bounds for…
An untouchable set in a projective plane is a set of points such that no line of the plane meets the set in exactly one point. Recently, H\'eger and Nagy (Avoiding Secants of Given Size in Finite Projective Planes, J. Combin. Des.…
We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also…
We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an…