{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T19:19:06Z","timestamp":1774984746416,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,24]],"date-time":"2025-01-24T00:00:00Z","timestamp":1737676800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In 1965, H. Retkin and E. Stein defined a symmetric point set as a set of points with the same intersection numbers. In this paper, we perform a detailed analysis of symmetric point sets of the finite projective space which shows that the class of symmetric sets is very broad including caps, two-character sets and transitive sets. We derive necessary conditions for the existence of such sets. Since the most studied sets are caps and two-character sets and not much seems to be known in the general case of sets, which are different from caps, with more than two intersection numbers, by using incidence-preserving group actions, symmetric point sets with few intersection numbers are provided. The results indicate that any finite projective space contains symmetric sets with few intersection numbers.<\/jats:p>","DOI":"10.3390\/sym17020179","type":"journal-article","created":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T04:59:10Z","timestamp":1737953950000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetric Point Sets with Few Intersection Numbers in PG(r,q)"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6109-6566","authenticated-orcid":false,"given":"Stefano","family":"Innamorati","sequence":"first","affiliation":[{"name":"Department of Industrial and Information Engineering and Economic, University of L\u2019Aquila, Piazzale Ernesto Pontieri, 1, I-67100 L\u2019Aquila, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,24]]},"reference":[{"key":"ref_1","unstructured":"Baker, C.A., and Batten, L.M. (1985). 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