Abstract
Two tantalizing invariants of a combinatorial code \({\mathcal {C}}\subseteq 2^{[n]}\) are \({{\,\mathrm{cdim}\,}}({\mathcal {C}})\) and \({{\,\mathrm{odim}\,}}({\mathcal {C}})\), the smallest dimension in which \({\mathcal {C}}\) can be realized by convex closed or open sets, respectively. Cruz, Giusti, Itskov, and Kronholm showed that for intersection complete codes \({\mathcal {C}}\) with \(m+1\) maximal codewords, \({{\,\mathrm{odim}\,}}({\mathcal {C}})\) and \({{\,\mathrm{cdim}\,}}({\mathcal {C}})\) are both bounded above by \(\max \{2,m\}\). Results of Lienkaemper, Shiu, and Woodstock imply that \({{\,\mathrm{odim}\,}}\) and \({{\,\mathrm{cdim}\,}}\) may differ, even for intersection complete codes. We add to the literature on open and closed embedding dimensions of intersection complete codes with the following results:
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If \({\mathcal {C}}\) is a simplicial complex, then \({{\,\mathrm{cdim}\,}}({\mathcal {C}}) = {{\,\mathrm{odim}\,}}({\mathcal {C}})\),
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If \({\mathcal {C}}\) is intersection complete, then \({{\,\mathrm{cdim}\,}}({\mathcal {C}})\le {{\,\mathrm{odim}\,}}({\mathcal {C}})\),
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If \({\mathcal {C}}\subseteq 2^{[n]}\) is intersection complete with \(n\ge 2\), then \({{\,\mathrm{cdim}\,}}({\mathcal {C}}) \le \min \{2d+1, n-1\}\), where d is the dimension of the simplicial complex of \({\mathcal {C}}\), and
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For each simplicial complex \(\Delta \subseteq 2^{[n]}\) with \(m\ge 2\) facets, the code \({\mathcal {S}}_\Delta \) \(:=(\Delta *(n+1)) \cup \{[n]\}\) \(\subseteq 2^{[n+1]}\) is intersection complete, has \(m+1\) maximal codewords, and satisfies \({{\,\mathrm{odim}\,}}({\mathcal {S}}_\Delta )=m\). In particular, for each \(n\ge 3\) there exists an intersection complete code \({\mathcal {C}}\subseteq 2^{[n]}\) with \({{\,\mathrm{odim}\,}}({\mathcal {C}}) = \left( {\begin{array}{c}n-1\\ \lfloor (n-1)/2\rfloor \end{array}}\right) \).
A key tool in our work is the study of sunflowers: arrangements of convex open sets in which the sets simultaneously meet in a central region, and nowhere else. We use Tverberg’s theorem to study the structure of “k-flexible" sunflowers, and consequently obtain new lower bounds on \({{\,\mathrm{odim}\,}}({\mathcal {C}})\) for intersection complete codes \({\mathcal {C}}\).
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References
Amenta, N., De Loera, J.A., Soberón, P.: Helly’s theorem: new variations and applications. In: Algebraic and Geometric Methods in Discrete Mathematics, volume 685 of Contemp. Math., pp. 55–95. American Mathematical Society, Providence, RI (2017)
Chen, A., Frick, F., Shiu, A.: Neural codes, decidability, and a new local obstruction to convexity. SIAM J. Appl. Algebra Geom. 3(1), 44–66 (2019)
Cruz, J., Giusti, C., Itskov, V., Kronholm, B.: On open and closed convex codes. Discrete Comput. Geom. 61, 247–270 (2016)
Curto, C., Gross, E., Jeffries, J., Morrison, K., Omar, M., Rosen, Z., Shiu, A., Youngs, N.: What makes a neural code convex? SIAM J. Appl. Algebra Geom. 1(1), 222–238 (2017)
Curto, C., Gross, E., Jeffries, J., Morrison, K., Rosen, Z., Shiu, A., Youngs, N.: Algebraic signatures of convex and non-convex codes. J. Pure Appl. Algebra 223(9), 3919–3940 (2019)
Curto, C., Itskov, V., Veliz-Cuba, A., Youngs, N.: The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes. Bull. Math. Biol. 75(9), 1571–1611 (2013)
Curto, C., Vera, R.: The Leray dimension of a convex code. arXiv e-prints: arXiv:1612.07797 (2016)
de Perez, A.R., Matusevich, L.F., Shiu, A.: Neural codes and the factor complex. Adv. Appl. Math 114, 101977 (2020)
Franke, M.K., Muthiah, S.: Every binary code can be realized by convex sets. Adv. Appl. Math. 99, 83–93 (2017)
Garcia, R., Garcia-Puente, L., Kruse, R., Liu, J., Miyata, D., Petersen, E., Phillipson, K., Shiu, A.: Gröbner bases of neural ideals. Int. J. Algebra Comput. 28(4), 553–571 (2018)
Goldrup, S.A., Phillipson, K.: Classification of open and closed convex codes on five neurons. Adv. Appl. Math. 112, 101948 (2020)
Gunturkun, S., Jeffries, J., Sun, J.: Polarization of neural rings. J. Algebra Appl. 19(8) (2019)
Itskov, V., Kunin, A., Rosen, Z.: Hyperplane neural codes and the polar complex. In: Nils, A., Baas, G.,Quick, Markus, S., Marius, T., Gunnar, E.C (eds.), Topological Data Analysis—The Abel Symposium, 2018, Abel Symposia, pp. 343–369. Springer (2020)
Jeffs, R.A.: Sunflowers of convex open sets. Adv. Appl. Math., 111, 101935 (2019)
Jeffs, R.A.: Morphisms of neural codes. SIAM J. Appl. Algebra Geom. 4, 99–122 (2020)
Jeffs, R.A.: Morphisms, Minors, and Minimal Obstructions to Convexity of Neural Codes. Ph.D. thesis, University of Washington, Seattle (2021). Available online at http://hdl.handle.net/1773/48062
Jeffs, R.A., Novik, I.: Convex union representability and convex codes. Int. Math. Res. Notices (2019)
Jeffs, R.A., Omar, M., Suaysom, N., Wachtel, A., Youngs, N.: Sparse neural codes and convexity. Involve J. Math. 12(5), 737–754 (2015)
Kunin, A., Lienkaemper, C., Rosen, Z.: Oriented matroids and combinatorial neural codes. arXiv e-prints: 2002.03542, arXiv:2002.03542 (2020)
Lienkaemper, C., Shiu, A., Woodstock, Z.: Obstructions to convexity in neural codes. Adv. Appl. Math. 85, 31–59 (2017)
Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer-Verlag, New York (2002)
Mulas, R., Tran, N.M.: Minimal embedding dimensions of connected neural codes. Algebraic Stat. 11(1), 99–106 (2020)
Schrijver, A.: Theory of integer and linear programming. Discrete Mathematics and Optimization, Wiley Interscience (1986)
Tancer, M.: d-representability of simplicial complexes of fixed dimension. J. Comput. Geom. 2(1), 183–188 (2011)
Tancer, M.: Intersection patterns of convex sets via simplicial complexes: a survey. In: Thirty essays on geometric graph theory, pp. 521–540. Springer, New York (2013)
Ziegler, G.M.: Lectures on polytopes. Graduate Texts in Mathematics, vol. 152. Springer-Verlag, New York (1995)
Acknowledgements
We would like to thank Florian Frick for raising the question of whether there exist open convex codes \({\mathcal {C}}\subseteq 2^{[n]}\) with \({{\,\mathrm{odim}\,}}({\mathcal {C}}) > n-1\), and for interesting discussions on this question. Isabella Novik provided detailed feedback on initial drafts of this paper, as well as helpful discussions on its mathematical content. Anne Shiu also provided thorough feedback on an initial draft of the paper. Finally, we would like to thank the anonymous referees for their comments, which greatly improved the presentation of our results.
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Jeffs’ research is supported by a graduate fellowship from NSF Grant DGE-1761124.