{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T16:10:35Z","timestamp":1762272635353,"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>We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollob\u00e1s, concerning colorings of $K_n$ where each color is well-represented. Let $\\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\\chi$ $\\varepsilon$-balanced if all color classes have $\\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\\varepsilon^r(H)$ such that for all $n\\geq R_\\varepsilon^r(H)$, all $\\varepsilon$-balanced colorings $\\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollob\u00e1s conjectured a simple characterization of $H$ for which $R_\\varepsilon^2(H)$ is finite, which was later proved by Cutler and Mont\u00e1gh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre.\u00a0<\/jats:p>","DOI":"10.37236\/8184","type":"journal-article","created":{"date-parts":[[2020,10,2]],"date-time":"2020-10-02T00:30:36Z","timestamp":1601598636000},"source":"Crossref","is-referenced-by-count":4,"title":["Finding Unavoidable Colorful Patterns in Multicolored Graphs"],"prefix":"10.37236","volume":"27","author":[{"given":"Matt","family":"Bowen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ander","family":"Lamaison","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alp","family":"M\u00fcyesser","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,10,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p4\/8185","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p4\/8185","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,2]],"date-time":"2020-10-02T00:30:36Z","timestamp":1601598636000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i4p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,2]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/8184","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,10,2]]},"article-number":"P4.4"}}