{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,20]],"date-time":"2025-09-20T08:45:30Z","timestamp":1758357930133,"version":"3.44.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Assume that $R_1,R_2,\\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides a positive answer in an asymptotic sense: in any set $S_1,\\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings.<\/jats:p>","DOI":"10.37236\/13267","type":"journal-article","created":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T14:17:48Z","timestamp":1758291468000},"source":"Crossref","is-referenced-by-count":0,"title":["Clique Covers of Complete Graphs and Piercing Multitrack Intervals"],"prefix":"10.37236","volume":"32","author":[{"given":"J\u00e1nos","family":"Bar\u00e1t","sequence":"first","affiliation":[]},{"given":"Andr\u00e1s","family":"Gy\u00e1rf\u00e1s","sequence":"additional","affiliation":[]},{"given":"G\u00e1bor N.","family":"S\u00e1rk\u00f6zy","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2025,9,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i3p45\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i3p45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T14:17:49Z","timestamp":1758291469000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i3p45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,19]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/13267","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,9,19]]},"article-number":"P3.45"}}