# The minrank of random graphs over arbitrary fields

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## Abstract

The minrank of a graph *G* on the set of vertices [*n*] over a field \(\mathbb{F}\) is the minimum possible rank of a matrix \(M \in \mathbb{F}^{{n}\times{n}}\) with nonzero diagonal entries such that *M*_{i,j} = 0 whenever *i* and *j* are distinct nonadjacent vertices of *G*. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph *G*(*n*, *p*) over any finite or infinite field, showing that for every field \(\mathbb{F}=\mathbb{F}(n)\) and every *p* = *p*(*n*) satisfying *n*^{-1} ≤ *p* ≤ 1 - *n*^{-0.99}, the minrank of *G* = *G*(*n*, *p*) over \(\mathbb{F}\) is \(\Theta(\frac{n {\rm{log}}(1/p)}{{\rm{log}} n})\) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev and Weinstein, who proved the above result for finite fields of size at most *n*^{O(1)}, with tools from linear algebra, including an estimate of Rónyai, Babai and Ganapathy for the number of zero-patterns of a sequence of polynomials.

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## Notes

### Acknowledgements

We thank Peter Nelson for telling us about his paper [16]. The research on this project was initiated during a joint research workshop of Tel Aviv University and the Free University of Berlin on Graph and Hypergraph Coloring Problems, held in Berlin in August 2018, and supported by a GIF grant number G-1347-304.6/2016. We would like to thank the German–Israeli Foundation (GIF) and both institutions for their support.

## References

- [1]N. Alon,
*The Shannon capacity of a union*, Combinatorica**18**(1998), 1998–301.MathSciNetCrossRefGoogle Scholar - [2]N. Alon,
*Lovász, vectors, graphs and codes*, manuscript https://www.tau.ac.il/~nogaa/PDFS/ll70.pdf. - [3]N. Alon and A. Kupavskii,
*Two notions of unit-distance graphs*, Journal of Combinatorual Theory. Series A**125**(2014), 2014–1.MathSciNetzbMATHGoogle Scholar - [4]Z. Bar-Yossef, Y. Birk, T. S. Jayram and T. Kol,
*Index coding with side information*, IEEE Transactions on Information Theory**57**(2011), 2011–1479.MathSciNetCrossRefGoogle Scholar - [5]B. Bollobás,
*The chromatic number of random graphs*, Combinatorica**8**(1988), 1988–49.MathSciNetCrossRefGoogle Scholar - [6]A. Golovnev, O. Regev and O. Weinstein,
*The minrank of random graphs*, in*Approximation, Randomization, and Combinatorial Optimization*.*Algorithms and Techniques*, Leibniz International Proceedings in Informatics, Vol. 81, Schloss Dagstuhl–Leibniz- Zentrum fuer Informatik, Dagstuhl, 2017, Art. no. 46.Google Scholar - [7]C. Grosu, F
_{p}*is locally like*C, Journal of the London Mathematical Society**89**(2014), 2014–724.CrossRefGoogle Scholar - [8]W. Haemers,
*An upper bound for the Shannon capacity of a graph*, in*Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978)*, Colloquia Mathematica Societatis János Bolyai, Vol. 25, North-Holland, Amsterdam–New York, 1981, pp. 267–272.Google Scholar - [9]I. Haviv,
*On minrank and forbidden subgraphs*, in*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques*, Leibniz International Proceedings in Informatics, Vol. 116, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, 2018, Art. no. 42.Google Scholar - [10]I. Haviv and M. Langberg,
*On linear index coding for random graphs*, in*2012 IEEE International Symposium on Information Theory Proceedings*, Institute of Electrical and Electronics Engineers, New York, 2012, pp. 2231–2235.CrossRefGoogle Scholar - [11]S. Janson, T. Łuczak and A. Ruciński,
*Random Graphs*, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.Google Scholar - [12]F. Juhász,
*The asymptotic behaviour of Lovász’ v function for random graphs*, Combinatorica**2**(1982), 1982–153.CrossRefGoogle Scholar - [13]D. E. Knuth,
*The sandwich theorem*, Electronic Journal of Combinatorics**1**(1994), Art. no. 1.Google Scholar - [14]L. Lovász,
*On the Shannon capacity of a graph*, IEEE Transactions on Information Theory**25**(1979), 1979–1.MathSciNetCrossRefGoogle Scholar - [15]E. Lubetzky and U. Stav,
*Non-linear index coding outperforming the linear optimum*, IEEE Transactions on Information Theory**55**(2009), 2009–3544.CrossRefGoogle Scholar - [16]P. Nelson,
*Almost all matroids are nonrepresentable*, Bulletin of the London Mathematical Society**50**(2018), 2018–245.MathSciNetCrossRefGoogle Scholar - [17]L. Rónyai, L. Babai and M. K. Ganapathy,
*On the number of zero-patterns of a sequence of polynomials*, Journal of the American Mathematical Society**14**(2001), 2001–717.MathSciNetCrossRefGoogle Scholar - [18]T. Tao,
*Rectification and the Lefschetz principle*, 14 March 2013, http://terrytao.wordpress.com/2013/03/14/rectification-and-the-lefschetz-principle. Google Scholar