Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Extending Arcs: An Elementary Proof. | Extending Arcs An Elementary Proof T. Alderson Department of Mathematical Sciences University of New Brunswick Saint John . Canada talderso@ Submitted Jan 19 2005 Accepted Apr 7 2005 Published Apr 28 2005 Mathematics Subject Classifications 51E21 51E15 Abstract In a finite projective plane K we consider two configuration conditions involving arcs in K and show via combinatorial means that they are equivalent. When the conditions hold we are able to obtain embeddability results for arcs all proofs being elementary. In particular when K PG 2 q with q even we provide short proofs of some well known embeddability results. 1 Introduction Let w be a projective plane of order n where n can be even or odd. An arc of size k or a k-arc in w is defined as a set of k points no three of which are collinear. If K is a k-arc and P is a point with K u P a k 1 -arc we say that P is an extending point of K. An arc is said to be complete if it possesses no extending points. A line is said to be a tangent resp. secant of an arc K if meets exactly one point resp. two points of K. In a plane of even resp. odd order an arc can have size at most n 2 resp. n 1 in which case it is an hyperoval resp. oval . For background on arcs in projective planes see 3 or 5 . Given a k-arc K in w we define a parameter Ỏ associated with K as follows k Ỏ n 2. We consider the following two conditions in w Condition A Every arc of size k n-24 is contained in a unique complete arc. Condition B If K is a complete k-arc then no point of w lies on as many as Ỏ 2 tangents of K. For k n 3 then Condition B is met trivially so the condition is one on complete arcs of reasonable size. It is well known that the classical planes PG 2 q where q is even The author acknowledges support from the . of Canada THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N6 1 meet Condition A 7 3 . Not all finite planes of even order satisfy Condition A we mention a class of counterexamples due to Menichetti 4 . Our first