{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T20:42:23Z","timestamp":1696538543453},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"Eckhoff proposed a combinatorial version of the classical Hadwiger\u2013Debrunner $(p,q)$-problems as follows. Let ${\\cal F}$ be a finite family of convex sets in the plane and\u00a0 let $m\\geqslant 1$ be an integer. If among every ${m+2\\choose 2}$ members of ${\\cal F}$ all but at most $m-1$ members have a common point, then there is a common point for all but at most $m-1$ members of ${\\cal F}$. The claim is an extension of Helly's theorem ($m=1$). The case $m=2$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemer\u00e9di and Petruska concerning $3$-uniform hypergraphs. This conjecture is still open in general; its\u00a0 solution for a few special cases answers Eckhoff's problem for $m=3,4$. A new proof for the case $m=2$ is also presented.<\/jats:p>","DOI":"10.37236\/9978","type":"journal-article","created":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T01:34:11Z","timestamp":1631842451000},"source":"Crossref","is-referenced-by-count":2,"title":["Eckhoff's Problem on Convex Sets in the Plane"],"prefix":"10.37236","volume":"28","author":[{"given":"Adam S.","family":"Jobson","sequence":"first","affiliation":[]},{"given":"Andr\u00e9 E.","family":"K\u00e9zdy","sequence":"additional","affiliation":[]},{"given":"Jen\u0151","family":"Lehel","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2021,8,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i3p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i3p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T01:34:11Z","timestamp":1631842451000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i3p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,27]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/9978","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,27]]}}}