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We partially fill this gap by proving that whenever<jats:inline-formula><jats:alternatives><jats:tex-math>$$v \\equiv 39$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>\u2261<\/mml:mo><mml:mn>39<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>(mod 72), or<jats:inline-formula><jats:alternatives><jats:tex-math>$$v \\equiv 4^e48 + 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>\u2261<\/mml:mo><mml:msup><mml:mn>4<\/mml:mn><mml:mi>e<\/mml:mi><\/mml:msup><mml:mn>48<\/mml:mn><mml:mo>+<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>(mod<jats:inline-formula><jats:alternatives><jats:tex-math>$$4^e96$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msup><mml:mn>4<\/mml:mn><mml:mi>e<\/mml:mi><\/mml:msup><mml:mn>96<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>) and<jats:inline-formula><jats:alternatives><jats:tex-math>$$e \\ge 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>e<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there exists a KTS on<jats:italic>v<\/jats:italic>points having at least<jats:inline-formula><jats:alternatives><jats:tex-math>$$v-3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group<jats:italic>G<\/jats:italic>acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of<jats:italic>G<\/jats:italic>, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.<\/jats:p>","DOI":"10.1007\/s10623-021-00952-x","type":"journal-article","created":{"date-parts":[[2021,10,7]],"date-time":"2021-10-07T07:17:58Z","timestamp":1633591078000},"page":"2725-2757","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class"],"prefix":"10.1007","volume":"89","author":[{"given":"Simona","family":"Bonvicini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1140-2251","authenticated-orcid":false,"given":"Marco","family":"Buratti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martino","family":"Garonzi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gloria","family":"Rinaldi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tommaso","family":"Traetta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,10,7]]},"reference":[{"key":"952_CR1","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1006\/jcta.1999.2969","volume":"88","author":"IJ Anderson","year":"1999","unstructured":"Anderson I.J., Finizio N.J., Leonard P.A.: New product theorems for $$Z$$-cyclic whist tournaments. 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