{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:09Z","timestamp":1753893849944,"version":"3.41.2"},"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>Given sets $\\mathcal{P}, \\mathcal{Q} \\subseteq \\mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\\mathcal{P} \\times \\mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \\leq n$. Elekes showed that $D(m, n) = O(\\sqrt{mn})$ when $m \\leqslant n^{1\/3}$. For $m \\geqslant n^{1\/3}$, we have the upper bound $D(m, n) = O(n\/\\sqrt{\\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \\Omega(\\sqrt{mn})$ when $m \\leqslant n^{1\/3}$. This is done by adapting Sz\u00e9kely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \\Omega(\\sqrt{mn}\/\\log n)$ when $m \\geqslant n^{1\/3}$.<\/jats:p>","DOI":"10.37236\/9687","type":"journal-article","created":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T23:54:33Z","timestamp":1637279673000},"source":"Crossref","is-referenced-by-count":0,"title":["On Bipartite Distinct Distances in the Plane"],"prefix":"10.37236","volume":"28","author":[{"given":"Surya","family":"Mathialagan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,11,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p33\/8456","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p33\/8456","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T23:54:33Z","timestamp":1637279673000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,19]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/9687","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,11,19]]},"article-number":"P4.33"}}