Abstract
This treatise deals with the isoperimetric problem in finite projective planes. We prove that certain sets, called (0,n,n+1)-sets, are solutions to this problem. This class of sets includes all the previously known solutions to the isoperimetric problem, as well as two new types of solutions which exist in every finite projective plane. We prove a characterization theorem for (0,n,n+1)-sets with many points. We solve the isoperimetric problem for large set size, and for q + 3 points if q is even. We find all the (0,n,n+1)-sets in planes of order at most 8 and develop techniques for proving that some (0,n,n+1)-sets in larger order planes do not exist. We solve the isoperimetric problem in the planes of order at most 7 (the solution was known only for planes of order at most 4), proving that nested solutions exist in these planes. We prove that no nested solutions exist in PG(2,8). We give examples of (0,2,3)-sets in planes of order 7, 8 and 16 which are new solutions to the isoperimetric problem not included in the infinite classes mentioned above, and we investigate Latin squares and Steiner triple systems associated with these examples.
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