The prime grid contains arbitrarily large empty polygons
-
Travis Dillon
Abstract
This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and Roldán-Pensado that the “prime grid” {(p, q) ∈ ℤ2 : p and q are prime} ⊆ ℝ2 contains empty polygons with arbitrarily many vertices. This implies that no Helly-type theorem is true for the prime grid.
Funding statement: This work was partially supported by a National Science Foundation Graduate Research Fellowship under Grant No. 2141064.
Acknowledgements
I gladly thank Pablo Soberón for first discussing this problem with me, Ashwin Sah for an insightful conversation about it, and Henry Cohn for reviewing a draft of this paper.
-
Communicated by: M. Henk
References
[1] G. Ambrus, M. Balko, N. Frankl, A. Jung, M. Naszódi, On Helly numbers of exponential lattices. European J. Combin. 116 (2024), Paper No. 103884, 17 pages. MR4666810 Zbl 1532.5200510.1016/j.ejc.2023.103884Search in Google Scholar
[2] N. Amenta, J. A. De Loera, P. Soberón, Helly’s theorem: new variations and applications. In: Algebraic and geometric methods in discrete mathematics, volume 685 of Contemp. Math., 55–95, Amer. Math. Soc. 2017. MR3625571 Zbl 1383.5200610.1090/conm/685/13718Search in Google Scholar
[3] S. Arun, T. Dillon, Variations on a theme of empty polytopes. Preprint 2024, arXiv 2409.07262v1Search in Google Scholar
[4] G. Averkov, R. Weismantel, Transversal numbers over subsets of linear spaces. Adv. Geom. 12 (2012), 19–28. MR2911157 Zbl 1250.5200610.1515/advgeom.2011.028Search in Google Scholar
[5] W. D. Banks, T. Freiberg, C. L. Turnage-Butterbaugh, Consecutive primes in tuples. Acta Arith. 167 (2015), 261–266. MR3316460 Zbl 1323.1107110.4064/aa167-3-4Search in Google Scholar
[6] I. Bárány, G. Kalai, Helly-type problems. Bull. Amer. Math. Soc. (N.S.) 59 (2022), 471–502. MR4478031 Zbl 1503.5200910.1090/bull/1753Search in Google Scholar
[7] D. E. Bell, A theorem concerning the integer lattice. Studies in Appl. Math. 56 (1976/77), 187–188. MR462617 Zbl 0388.9005110.1002/sapm1977562187Search in Google Scholar
[8] Z. Chase, A random analogue of Gilbreath’s conjecture. Math. Ann. 388 (2024), 2611–2625. MR4705747 Zbl 0780805810.1007/s00208-023-02579-wSearch in Google Scholar
[9] L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives. In: Proc. Sympos. Pure Math., Vol. VII, 101–180, Amer. Math. Soc. 1963. MR157289 Zbl 0132.1740110.1090/pspum/007/0157289Search in Google Scholar
[10] J. A. De Loera, R. N. La Haye, D. Oliveros, E. Roldán-Pensado, Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s theorem. Adv. Geom. 17 (2017), 473–482. MR3714450 Zbl 1431.5201110.1515/advgeom-2017-0028Search in Google Scholar
[11] T. Dillon, Discrete quantitative Helly-type theorems with boxes. Adv. in Appl. Math. 129 (2021), Paper No. 102217, 17 pages. MR4255344 Zbl 1509.5201410.1016/j.aam.2021.102217Search in Google Scholar
[12] J.-P. Doignon, Convexity in cristallographical lattices. J. Geom. 3 (1973), 71–85. MR387090 Zbl 0245.5200410.1007/BF01949705Search in Google Scholar
[13] P. Erdös, P. Turán, On some new questions on the distribution of prime numbers. Bull. Amer. Math. Soc. 54 (1948), 371–378. MR24460 Zbl 0032.2690210.1090/S0002-9904-1948-09010-3Search in Google Scholar
[14] E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Dtsch. Math.-Ver. 32 (1923), 175–176. JFM 49.0534.02Search in Google Scholar
[15] A. J. Hoffman, Binding constraints and Helly numbers. In: Second International Conference on Combinatorial Mathematics (New York, 1978), volume 319 of Ann. New York Acad. Sci., 284–288, New York Acad. Sci., New York 1979. MR556036 Zbl 0489.5201110.1111/j.1749-6632.1979.tb32803.xSearch in Google Scholar
[16] C. A. J. Hurkens, Blowing up convex sets in the plane. Linear Algebra Appl. 134 (1990), 121–128. MR1060014 Zbl 0708.5200210.1016/0024-3795(90)90010-ASearch in Google Scholar
[17] A. Ya. Khinchin, A quantitative formulation of the approximation theory of Kronecker. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 113–122. MR24925 Zbl 0030.02002Search in Google Scholar
[18] J. Maynard, Small gaps between primes. Ann. of Math. (2) 181 (2015), 383–413. MR3272929 Zbl 1306.1107310.4007/annals.2015.181.1.7Search in Google Scholar
[19] J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann. 83 (1921), 113–115. MR1512002 JFM 48.0834.0410.1007/BF01464231Search in Google Scholar
[20] H. E. Scarf, An observation on the structure of production sets with indivisibilities. Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 3637–3641. MR452678 Zbl 0381.9008110.1073/pnas.74.9.3637Search in Google Scholar PubMed PubMed Central
[21] K. B. Summers, The Helly number of the prime-coordinate point set. Bachelor’s thesis, University of California, 2015.Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
