{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:25:57Z","timestamp":1760059557080,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T00:00:00Z","timestamp":1750464000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The Ramsey number\u00a0R(F)\u00a0of a graph F without isolated vertices is the smallest positive integer n such that every red\u2013blue coloring of\u00a0Kn\u00a0produces a subgraph isomorphic to F all of whose edges are colored the same. Let\u00a0F\u00a0be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number\u00a0VRt(F)\u00a0is the smallest positive integer n such that every red\u2013blue coloring of the complete graph\u00a0Kn\u00a0of order n results in at least t pairwise vertex-disjoint monochromatic graphs in\u00a0F; while the edge-disjoint Ramsey number\u00a0ERt(F)\u00a0is the smallest positive integer n such that every red\u2013blue coloring of\u00a0Kn\u00a0produces at least t pairwise edge-disjoint monochromatic graphs in\u00a0F. If\u00a0t=1\u00a0and\u00a0F\u00a0consists of a single graph F, then\u00a0VR1(F)=ER1(F)=R(F)\u00a0is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for\u00a0VRt(F)\u00a0and\u00a0ERt(F)\u00a0are established for sets\u00a0F\u00a0of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of\u00a0VRt(F)\u00a0and\u00a0ERt(F)\u00a0for sets\u00a0F\u00a0of graphs of size 2 or 3 without isolated vertices. The exact values of\u00a0VRt(F)\u00a0are determined for all such sets\u00a0F\u00a0and all integers\u00a0t\u22652. The exact values of\u00a0ERt(F)\u00a0of certain such sets\u00a0F\u00a0with prescribed conditions for all integers\u00a0t\u22652\u00a0are determined. For some special sets\u00a0F\u00a0of graphs of size 2 or 3 without isolated vertices, the exact values of\u00a0ERt(F)\u00a0are determined for\u00a02\u2264t\u22644. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general.<\/jats:p>","DOI":"10.3390\/axioms14070486","type":"journal-article","created":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T07:42:34Z","timestamp":1750664554000},"page":"486","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Vertex-Disjoint and Edge-Disjoint Ramsey Numbers of a Set of Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Emma","family":"Jent","sequence":"first","affiliation":[{"name":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA"}]},{"given":"Ping","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,21]]},"reference":[{"key":"ref_1","unstructured":"Graham, R.L., Rothschild, B.L., and Spencer, J.H. 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