Finite projective planes with a large abelian group

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

Chapter 4 - Projective Planes

The matching number \(\alpha '(H)\) of a hypergraph H is the size of a largest matching in H, where a matching is a set of pairwise disjoint edges in H. A dominating set in H is a subset D of vertices of H such that for every \(v\in V(H)\setminus D\) there exists \(u\in D\) such that u and v lie in an edge of H, and the domination number of H, denoted by \(\gamma (H)\), is the minimum cardinality of a dominating set in H. It was shown that a Ryser-like inequality \(\gamma (H)\le (r-1)\alpha '(H)\) holds for hypergraphs H of rank r. In particular, for intersecting hypergraphs H of rank r, \(\gamma (H)\le r-1\), since \(\alpha '(H)=1\). The linear intersecting hypergraphs of rank \(2\le r\le 4\) achieving the equality \(\gamma (H)=r-1\) have been characterized. In this paper we show that all the 5-uniform linear intersecting hypergraphs H with equality \(\gamma (H)=r-1\) are generated by the finite projective plane of order three.

We find necessary conditions that should be imposed on a number n for the existence of an arbitrary configuration in at least one finite projective plane of order n.

Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 32h). Also, we show that these blocking sets of PG(2, q2), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 32h) gives rise to a family of blocking semiovals in the André plane as well.

We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.

ABSTRACTLet be a prime power and be a natural number. What are the possible cardinalities of point sets in a projective plane of order , which do not intersect any line at exactly points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values for , provided that is not close to the extremal values 0 or . Moreover, using polynomial techniques we show that for every prescribed list of numbers , there exists a point set with the following property: holds for the th line , .

Triangle-free induced subgraphs of the unitary polarity graph

This research is concerned with the study of the projective plane over a finite field . The main purpose is finding partitions of the projective line PG( ) and the projective plane PG( ) , in addition to embedding PG(1, ) into PG( ) and PG( ) into PG( ). Clearly, the orbits of PG( ) are found, along with the cross-ratio for each orbit. As for PG( ), 13 partitions were found on PG( ) each partition being classified in terms of the degree of its arc, length, its own code, as well as its error correcting. The last main aim is to classify the group actions on PG( ).

CHAPTER 6 - COORDINATIZATION

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal nonpercolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

In the late 1950’s, B. Segre introduced the fundamental notion of arcs and complete arcs [48,49]. An arc in a finite projective plane is a set of points with no three on a line and it is complete if cannot be extended without violating this property. Given a projective plane $${\user1{P}}$$, determining $$n{\left( {\user1{P}} \right)}$$, the size of its smallest complete arc, has been a major open question in finite geometry for several decades. Assume that $${\user1{P}}$$ has order q, it was shown by Lunelli and Sce [41], more than 40 years ago, that $${\left( {\user1{P}} \right)} \geqslant {\sqrt {2q} }$$. Apart from this bound, practically nothing was known about $$n{\left( {\user1{P}} \right)}$$, except for the case $${\user1{P}}$$ is the Galois plane. For this case, the best upper bound, prior to this paper, was O(q3/4) obtained by Szőnyi using the properties of the Galois field GF(q).In this paper, we prove that $$n{\left( {\user1{P}} \right)} \leqslant {\sqrt q }{\kern 1pt} \log^{c} q$$ for any projective plane $${\user1{P}}$$ of order q, where c is a universal constant. Together with Lunelli-Sce’s lower bound, our result determines $$n{\left( {\user1{P}} \right)}$$ up to a polylogarithmic factor. Our proof uses a probabilistic method known as the dynamic random construction or Rodl’s nibble. The proof also gives a quick randomized algorithm which produces a small complete arc with high probability.The key ingredient of our proof is a new concentration result, which applies for non-Lipschitz functions and is of independent interest.

Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties

We present a fairly elementary, self-contained proof of the nonexistence of a finite projective plane of order $6$. Our approach is motivated by theory of binary codes but does not appeal to it directly.

We study a construction of the Mathieu group M 12 using a game reminiscent of Loyd's "15-puzzle." The elements of M 12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M 13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M 12 and M 13. We develop these results, and extend them to the double covers and automorphism groups of M 12 and M 13, using the ternary Golay code and 12 × 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.