article published yes Nemanja Draganić author Kalina H Petrova author 554ff4e4-f325-11ee-b0c4-a10dbd523381 MaKw department IST-BRIDGE: International postdoctoral program project The size-Ramsey number r^(H) of a graph H is the smallest number of edges a (host) graph G can have, such that for any red/blue colouring of G, there is a monochromatic copy of H in G. Recently, Conlon, Nenadov and Trujić showed that if H is a graph on n vertices and maximum degree three, then r^(H)=O(n8/5), improving upon the upper bound of n5/3+o(1) by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper we show that r^(H)≤n3/2+o(1). While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods. https://research-explorer.ista.ac.at/download/19418/19427/2025_JournLondonMath_Draganic.pdf application/pdfno Wiley2025 eng 0024-6107 1469-7750 2207.05048 00145064540001910.1112/jlms.70116 1113 Draganić, Nemanja, and Kalina H. Petrova. “Size‐Ramsey Numbers of Graphs with Maximum Degree Three.” <i>Journal of the London Mathematical Society</i>, vol. 111, no. 3, e70116, Wiley, 2025, doi:<a href="https://doi.org/10.1112/jlms.70116">10.1112/jlms.70116</a>. Draganić N, Petrova KH. 2025. Size‐Ramsey numbers of graphs with maximum degree three. Journal of the London Mathematical Society. 111(3), e70116. Draganić N, Petrova KH. Size‐Ramsey numbers of graphs with maximum degree three. <i>Journal of the London Mathematical Society</i>. 2025;111(3). doi:<a href="https://doi.org/10.1112/jlms.70116">10.1112/jlms.70116</a> Draganić, N., &#38; Petrova, K. H. (2025). Size‐Ramsey numbers of graphs with maximum degree three. <i>Journal of the London Mathematical Society</i>. Wiley. <a href="https://doi.org/10.1112/jlms.70116">https://doi.org/10.1112/jlms.70116</a> N. Draganić, K.H. Petrova, Journal of the London Mathematical Society 111 (2025). N. Draganić and K. H. Petrova, “Size‐Ramsey numbers of graphs with maximum degree three,” <i>Journal of the London Mathematical Society</i>, vol. 111, no. 3. Wiley, 2025. Draganić, Nemanja, and Kalina H Petrova. “Size‐Ramsey Numbers of Graphs with Maximum Degree Three.” <i>Journal of the London Mathematical Society</i>. Wiley, 2025. <a href="https://doi.org/10.1112/jlms.70116">https://doi.org/10.1112/jlms.70116</a>. 194182025-03-19T09:03:37Z2025-09-30T11:05:07Z