{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:28:31Z","timestamp":1772119711248,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $\\Pi = (P,L,I)$ denote a rank two geometry. In this paper, we are interested in the largest value of $|X||Y|$\u00a0where $X \\subset P$ and $Y \\subset L$ are sets such that $(X \\times Y) \\cap I = \\emptyset$. Let $\\alpha(\\Pi)$ denote this value.\u00a0We concentrate on the case where $P$ is the point set of $\\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\\mathsf{PG}(n,q)$.\u00a0In the case that $\\Pi$ is the projective plane $\\mathsf{PG}(2,q)$, where $P$ is the set of points and $L$ is the set of lines of the projective plane, Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine $\\alpha(\\mathsf{PG}(2,q))$ when $q$ is an even power of $2$. Therefore, in those cases,\\[ \\alpha(\\Pi) = q(q - \\sqrt{q} + 1)^2.\\]\u00a0We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in\u00a0generalized polygons. More generally, if $P$ is the point set of $\\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\\mathsf{PG}(n,q)$, where $1 \\leq k \\leq n - 1$,\u00a0and $\\Pi_q = (P,L,I)$, then we show as $q \\rightarrow \\infty$ that\u00a0\\[ \\frac{1}{4}q^{(k + 2)(n - k)} \\lesssim \\alpha(\\Pi) \\lesssim q^{(k + 2)(n - k)}.\\]\u00a0The upper bounds are proved by combinatorial and spectral techniques. This leaves the open question as to the smallest possible value of $\\alpha(\\Pi)$ for each value of $k$. We prove that if for each $N \\in \\mathbb N$, $\\Pi_N$ is a partial linear space with $N$ points and $N$ lines, then $\\alpha(\\Pi_N) \\gtrsim \\frac{1}{e}N^{3\/2}$ as $N \\rightarrow \\infty$.<\/jats:p>","DOI":"10.37236\/2831","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:11:32Z","timestamp":1578712292000},"source":"Crossref","is-referenced-by-count":16,"title":["Large Incidence-free Sets in Geometries"],"prefix":"10.37236","volume":"19","author":[{"given":"Stefaan","family":"De Winter","sequence":"first","affiliation":[]},{"given":"Jeroen","family":"Schillewaert","sequence":"additional","affiliation":[]},{"given":"Jacques","family":"Verstraete","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,11,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i4p24\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i4p24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:22:59Z","timestamp":1579299779000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i4p24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,8]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2012,10,18]]}},"URL":"https:\/\/doi.org\/10.37236\/2831","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,11,8]]},"article-number":"P24"}}