Abstract
Given a k-uniform hypergraph ℋ and sufficiently large m ≫ m0(ℋ), we show that an m-element set I ⊆ V(ℋ), chosen uniformly at random, with probability 1 − e−ω(m) is either not independent or is contained in an almost-independent set in ℋ which, crucially, can be constructed from carefully chosen o(m) vertices of I. As a corollary, this implies that if the largest almost-independent set in ℋ is of size o(v(ℋ)) then I itself is an independent set with probability e−ω(m). More generally, I is very likely to inherit structural properties of almost-independent sets in ℋ.
The value m0(ℋ) coincides with that for which Janson’s inequality gives that I is independent with probability at most \({e^{- \Theta ({m_0})}}\). On the one hand, our result is a significant strengthening of Janson’s inequality in the range m ≫ m0. On the other hand, it can be seen as a probabilistic variant of hypergraph container theorems, developed by Balogh, Morris and Samotij and, independently, by Saxton and Thomason. While being strictly weaker than the original container theorems in the sense that it does not apply to all independent sets of size m, it is nonetheless sufficient for many applications and admits a short proof using probabilistic ideas.
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References
N. Alon, J. Balogh, R. Morris and W. Samotij, Counting sum-free sets in abelian groups, Israel Journal of Mathematics 199 (2014), 309–344.
N. Alon and J. H. Spencer, The Probabilistic Method, Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Hoboken, NJ, 2016.
J. Balogh, H. Liu and M. Sharifzadeh, The number of subsets of integers with no k-term arithmetic progression, International Mathematics Research Notices 2017 (2017), 6168–6186.
J. Balogh, R. Morris and W. Samotij, Independent sets in hypergraphs, Journal of the American Mathematical Society 28 (2015), 669–709.
J. Balogh, R. Morris and W. Samotij, The method of hypergraph containers, in Proceedings of the International Congress of Mathematicians—Rio De Janeiro 2018, Vol. IV. Invited Lectures, World Scientific, Hackensack, NJ, 2018, pp. 3059–3092.
J. Balogh and Š. Petříčková, The number of the maximal triangle-free graphs, Bulletin of the London Mathematical Society 46 (2014), 1003–1006.
J. Balogh and W. Samotij, An efficient container lemma, Discrete Analysis 2020 (2020), Article no. 17.
J. Balogh and J. Solymosi, On the number of points in general position in the plane, Discrete Analysis 2018 (2018), Article no. 16.
A. Bernshteyn, M. Delcourt, H. Towsner and A. Tserunyan, A short nonalgorithmic proof of the containers theorem for hypergraphs, Proceedings of the American Mathematical Society 147 (2019), 1739–1749.
D. Conlon, W. T. Gowers, W. Samotij and M. Schacht, On the KLR conjecture in random graphs, Israel Journal of Mathematics 203 (2014), 535–580.
D. Conlon and J. Fox, Graph removal lemmas, in Surveys in Combinatorics 2013, London Mathematical Society Lecture Note Series, Vol. 409, Cambridge University Press, Cambridge, 2013, pp. 1–49.
D. Conlon and W. T. Gowers, Combinatorial theorems in sparse random sets, Annals of Mathematics 184 (2016), 367–454.
A. Ferber, G. McKinley and W. Samotij, Supersaturated sparse graphs and hypergraphs, International Mathematics Research Notices 2020 (2020), 378–402.
Y. Kohayakawa, T. Łuczak and V. Rödl, On K4-free subgraphs of random graphs, Combinatorica 17 (1997), 173–213.
R. Morris and D. Saxton, The number of C2ℓ-free graphs, Advances in Mathematics 298 (2016), 534–580.
R. Nenadov, Small subsets without k-term arithmetic progressions, https://arxiv.org/abs/2109.02964.
R. Nenadov, A new proof of the KŁR conjecture, Advances in Mathematics 406 (2022), Article no. 108518.
V. Rödl and A. Ruciński, Threshold functions for Ramsey properties, Journal of the American Mathematical Society 8 (1995), 917–942.
W. Samotij, Counting independent sets in graphs, European Journal of Combinatorics 48 (2015), 5–18.
D. Saxton and A. Thomason, Hypergraph containers, Inventiones Mathematicae 201 (2015), 925–992.
D. Saxton and A. Thomason, Simple containers for simple hypergraphs, Combinatorics, Probability and Computing 25 (2016), 448–459.
M. Schacht, Extremal results for random discrete structures, Annals of Mathematics 184 (2016), 333–365.
E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199–245.
Acknowledgment
The author would like to thank Miloš Trujić and Andrew Thomason for comments on the early version of the manuscript. The author is also indebted to the anonymous referee for many helpful suggestions and for spotting a subtle (but serious) mistake in the setup of Theorem 1.1, which manifested itself in the proof of Lemma 2.2.
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Nenadov, R. Probabilistic hypergraph containers. Isr. J. Math. 261, 879–897 (2024). https://doi.org/10.1007/s11856-023-2602-9
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DOI: https://doi.org/10.1007/s11856-023-2602-9