article published yes Jacob Fox author János Pach author E62E3130-B088-11EA-B919-BF823C25FEA4 Andrew Suk author HeEd department In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension. Springer Nature2021 eng Computational MathematicsDiscrete Mathematics and Combinatorics 0209-9683 1439-6912 1912.0234210.1007/s00493-021-4530-9 416803-813 J. Fox, J. Pach, A. Suk, Combinatorica 41 (2021) 803–813. Fox, Jacob, János Pach, and Andrew Suk. “Bounded VC-Dimension Implies the Schur-Erdős Conjecture.” <i>Combinatorica</i>. Springer Nature, 2021. <a href="https://doi.org/10.1007/s00493-021-4530-9">https://doi.org/10.1007/s00493-021-4530-9</a>. Fox J, Pach J, Suk A. 2021. Bounded VC-dimension implies the Schur-Erdős conjecture. Combinatorica. 41(6), 803–813. Fox, J., Pach, J., &#38; Suk, A. (2021). Bounded VC-dimension implies the Schur-Erdős conjecture. <i>Combinatorica</i>. Springer Nature. <a href="https://doi.org/10.1007/s00493-021-4530-9">https://doi.org/10.1007/s00493-021-4530-9</a> Fox J, Pach J, Suk A. Bounded VC-dimension implies the Schur-Erdős conjecture. <i>Combinatorica</i>. 2021;41(6):803-813. doi:<a href="https://doi.org/10.1007/s00493-021-4530-9">10.1007/s00493-021-4530-9</a> Fox, Jacob, et al. “Bounded VC-Dimension Implies the Schur-Erdős Conjecture.” <i>Combinatorica</i>, vol. 41, no. 6, Springer Nature, 2021, pp. 803–13, doi:<a href="https://doi.org/10.1007/s00493-021-4530-9">10.1007/s00493-021-4530-9</a>. J. Fox, J. Pach, and A. Suk, “Bounded VC-dimension implies the Schur-Erdős conjecture,” <i>Combinatorica</i>, vol. 41, no. 6. Springer Nature, pp. 803–813, 2021. 152752024-04-03T07:59:57Z2024-04-09T10:40:08Z