[{"intvolume":"       348","abstract":[{"text":"For a field F and integers d and k, a set A ⊆ Fd is called k-nearly orthogonal if its\r\nmembers are non-self-orthogonal and every k + 1 vectors of A include an orthogonal pair.\r\nWe prove that for every prime p there exists some δ = δ(p)> 0, such that for every field\r\nF of characteristic p and for all integers k ≥ 2 and d ≥ k, there exists a k-nearly orthogonal\r\nset of at least dδ·k/ logk vectors of Fd. The size of the set is optimal up to the logk term\r\nin the exponent. We further prove two extensions of this result. In the first, we provide a\r\nlarge set A of non-self-orthogonal vectors of Fd such that for every two subsets of A of\r\nsize k+1 each, some vector of one of the subsets is orthogonal to some vector of the other.\r\nIn the second extension, every k + 1 vectors of the produced set A include ℓ + 1 pairwise\r\northogonal vectors for an arbitrary fixed integer 1 ≤ ℓ ≤ k. The proofs involve probabilistic\r\nand spectral arguments and the hypergraph container method","lang":"eng"}],"publication":"Discrete Mathematics","OA_place":"repository","title":"Larger nearly orthogonal sets over finite fields","publication_status":"published","OA_type":"green","day":"01","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2404.01057 "}],"language":[{"iso":"eng"}],"author":[{"full_name":"Haviv, Ishay","last_name":"Haviv","first_name":"Ishay"},{"last_name":"Mattheus","first_name":"Sam","full_name":"Mattheus, Sam"},{"last_name":"Milojević","first_name":"Aleksa","full_name":"Milojević, Aleksa"},{"full_name":"Wigderson, Yuval","last_name":"Wigderson","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","first_name":"Yuval"}],"article_processing_charge":"No","citation":{"mla":"Haviv, Ishay, et al. “Larger Nearly Orthogonal Sets over Finite Fields.” <i>Discrete Mathematics</i>, vol. 348, no. 4, 114373, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.disc.2024.114373\">10.1016/j.disc.2024.114373</a>.","ieee":"I. Haviv, S. Mattheus, A. Milojević, and Y. Wigderson, “Larger nearly orthogonal sets over finite fields,” <i>Discrete Mathematics</i>, vol. 348, no. 4. Elsevier, 2025.","short":"I. Haviv, S. Mattheus, A. Milojević, Y. Wigderson, Discrete Mathematics 348 (2025).","apa":"Haviv, I., Mattheus, S., Milojević, A., &#38; Wigderson, Y. (2025). Larger nearly orthogonal sets over finite fields. <i>Discrete Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.disc.2024.114373\">https://doi.org/10.1016/j.disc.2024.114373</a>","ama":"Haviv I, Mattheus S, Milojević A, Wigderson Y. Larger nearly orthogonal sets over finite fields. <i>Discrete Mathematics</i>. 2025;348(4). doi:<a href=\"https://doi.org/10.1016/j.disc.2024.114373\">10.1016/j.disc.2024.114373</a>","chicago":"Haviv, Ishay, Sam Mattheus, Aleksa Milojević, and Yuval Wigderson. “Larger Nearly Orthogonal Sets over Finite Fields.” <i>Discrete Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.disc.2024.114373\">https://doi.org/10.1016/j.disc.2024.114373</a>.","ista":"Haviv I, Mattheus S, Milojević A, Wigderson Y. 2025. Larger nearly orthogonal sets over finite fields. Discrete Mathematics. 348(4), 114373."},"oa":1,"arxiv":1,"publication_identifier":{"issn":["0012-365X"]},"month":"04","scopus_import":"1","volume":348,"oa_version":"Preprint","_id":"22163","article_number":"114373","das_tickbox":"1","doi":"10.1016/j.disc.2024.114373","external_id":{"arxiv":["2404.01057"]},"extern":"1","date_updated":"2026-07-14T08:17:17Z","date_published":"2025-04-01T00:00:00Z","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-06-29T10:53:05Z","keyword":["Nearly orthogonal sets","Ramsey theory","Finite fields"],"publisher":"Elsevier","type":"journal_article","article_type":"original","issue":"4","year":"2025"},{"external_id":{"arxiv":["2110.14483"]},"extern":"1","date_updated":"2026-07-14T08:27:40Z","mathsc":["05C55","05D10"],"doi":"10.1017/s0963548322000360","oa_version":"Preprint","_id":"22165","issue":"3","year":"2023","publisher":"Cambridge University Press","article_type":"original","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-06-29T10:53:47Z","keyword":["Ramsey theory","book graphs","Ramsey goodness"],"date_published":"2023-05-01T00:00:00Z","status":"public","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2110.14483","open_access":"1"}],"day":"01","quality_controlled":"1","OA_type":"green","intvolume":"        32","abstract":[{"text":"The book graph 𝐵(𝑘)\r\n𝑛 consists of 𝑛 copies of 𝐾𝑘+1 joined along a common 𝐾𝑘. In the prequel to this paper, we studied the diagonal Ramsey number 𝑟⁡(𝐵(𝑘)\r\n𝑛,𝐵(𝑘)\r\n𝑛). Here we consider the natural off-diagonal variant 𝑟⁡(𝐵(𝑘)\r\n𝑐⁢𝑛,𝐵(𝑘)\r\n𝑛) for fixed 𝑐 ∈(0,1]. In this more general setting, we show that an interesting dichotomy emerges: for very small 𝑐, a simple 𝑘-partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being 𝑘-partite, while, for 𝑐 bounded away from 0, random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of 𝑐.\r\n\r\n","lang":"eng"}],"OA_place":"repository","publication":"Combinatorics, Probability and Computing","publication_status":"published","title":"Off-diagonal book Ramsey numbers","scopus_import":"1","month":"05","volume":32,"publication_identifier":{"eissn":["1469-2163"],"issn":["0963-5483"]},"oa":1,"arxiv":1,"language":[{"iso":"eng"}],"author":[{"full_name":"Conlon, David","last_name":"Conlon","first_name":"David"},{"first_name":"Jacob","last_name":"Fox","full_name":"Fox, Jacob"},{"first_name":"Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","last_name":"Wigderson","full_name":"Wigderson, Yuval"}],"article_processing_charge":"No","citation":{"ieee":"D. Conlon, J. Fox, and Y. Wigderson, “Off-diagonal book Ramsey numbers,” <i>Combinatorics, Probability and Computing</i>, vol. 32, no. 3. Cambridge University Press, pp. 516–545, 2023.","mla":"Conlon, David, et al. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics, Probability and Computing</i>, vol. 32, no. 3, Cambridge University Press, 2023, pp. 516–45, doi:<a href=\"https://doi.org/10.1017/s0963548322000360\">10.1017/s0963548322000360</a>.","short":"D. Conlon, J. Fox, Y. Wigderson, Combinatorics, Probability and Computing 32 (2023) 516–545.","chicago":"Conlon, David, Jacob Fox, and Yuval Wigderson. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics, Probability and Computing</i>. Cambridge University Press, 2023. <a href=\"https://doi.org/10.1017/s0963548322000360\">https://doi.org/10.1017/s0963548322000360</a>.","ista":"Conlon D, Fox J, Wigderson Y. 2023. Off-diagonal book Ramsey numbers. Combinatorics, Probability and Computing. 32(3), 516–545.","ama":"Conlon D, Fox J, Wigderson Y. Off-diagonal book Ramsey numbers. <i>Combinatorics, Probability and Computing</i>. 2023;32(3):516-545. doi:<a href=\"https://doi.org/10.1017/s0963548322000360\">10.1017/s0963548322000360</a>","apa":"Conlon, D., Fox, J., &#38; Wigderson, Y. (2023). Off-diagonal book Ramsey numbers. <i>Combinatorics, Probability and Computing</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0963548322000360\">https://doi.org/10.1017/s0963548322000360</a>"},"page":"516-545"}]