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We denote by \n \n $$r_k(\\mathbb {F}_p^n)$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n p<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case \n \n $$k=p$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n =<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and therefore sets containing no full line. A\u00a0trivial lower bound \n \n $$r_p(\\mathbb {F}_p^n)\\ge (p-1)^n$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n p<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n \n \n (<\/mml:mo>\n p<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is achieved by a hypercube of side length \n \n $$p-1$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and it is known that equality holds for \n \n $$n\\in \\{1,2\\}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We will however show that \n \n $$r_p(\\mathbb {F}_p^3)\\ge (p-1)^3+p-2\\sqrt{p}$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n p<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n \n \n (<\/mml:mo>\n p<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n p<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n \n p<\/mml:mi>\n <\/mml:msqrt>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, which is the first improvement in the three-dimensional case that is increasing in \n \n $$p$$<\/jats:tex-math>\n \n p<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We will also give the upper bound \n \n $$r_p(\\mathbb {F}_p^{3})\\le p^3-2p^2-(\\sqrt{2}-1)p+2$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n p<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2264<\/mml:mo>\n \n p<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n -<\/mml:mo>\n 2<\/mml:mn>\n \n p<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n -<\/mml:mo>\n \n (<\/mml:mo>\n \n 2<\/mml:mn>\n <\/mml:msqrt>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n p<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as well as generalizations for higher dimensions. Finally, we present some bounds for individual \n \n $$p$$<\/jats:tex-math>\n \n p<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, in particular \n \n $$r_5(\\mathbb {F}_5^{3})\\ge 70$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n 5<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n 5<\/mml:mn>\n 3<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n 70<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$r_7(\\mathbb {F}_7^{3})\\ge 225$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n 7<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n 7<\/mml:mn>\n 3<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n 225<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> which can be used to give the asymptotic lower bound \n \n $$4.121^n$$<\/jats:tex-math>\n \n \n 4<\/mml:mn>\n .<\/mml:mo>\n \n 121<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for \n \n $$r_5(\\mathbb {F}_5^{n})$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n 5<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n 5<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$6.082^n$$<\/jats:tex-math>\n \n \n 6<\/mml:mn>\n .<\/mml:mo>\n \n 082<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for \n \n $$r_7(\\mathbb {F}_7^{n})$$<\/jats:tex-math>\n \n \n \n r<\/mml:mi>\n 7<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n F<\/mml:mi>\n 7<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10998-024-00617-x","type":"journal-article","created":{"date-parts":[[2025,1,4]],"date-time":"2025-01-04T06:42:20Z","timestamp":1735972940000},"page":"7-21","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Maximal line-free sets in $$\\mathbb {F}_p^n$$"],"prefix":"10.1007","volume":"90","author":[{"given":"Christian","family":"Elsholtz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jakob","family":"F\u00fchrer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erik","family":"F\u00fcredi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benedek","family":"Kov\u00e1cs","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P\u00e9ter P\u00e1l","family":"Pach","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D\u00e1niel G\u00e1bor","family":"Simon","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N\u00f3ra","family":"Velich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,4]]},"reference":[{"key":"617_CR1","unstructured":"A. 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