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Uniform hypergraphs containing no grids. (English) Zbl 1278.05161
Summary: A hypergraph is called an $$r \times r$$ grid if it is isomorphic to a pattern of $$r$$ horizontal and $$r$$ vertical lines, i.e., a family of sets $$\{A_1, \dots, A_r, B_1, \dots, B_r\}$$ such that $$A_i\cap A_j=B_i\cap B_j=\emptyset$$ for $$1\leq i<j\leq r$$ and $$|A_i\cap B_j|=1$$ for $$1\leq i,j\leq r$$. Three sets $$C_1,C_2,C_3$$ form a triangle if they pairwise intersect in three distinct singletons, $$|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1$$, $$C_1\cap C_2\neq C_1\cap C_3$$. A hypergraph is linear, if $$|E\cap F|\leq 1$$ holds for every pair of edges $$E\neq F$$. In this paper we construct large linear $$r$$-hypergraphs which contain no grids. Moreover, a similar construction gives large linear $$r$$-hypergraphs which contain neither grids nor triangles. For $$r\geq 4$$ our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs.

##### MSC:
 05C65 Hypergraphs 05C42 Density (toughness, etc.) 05D05 Extremal set theory 11B25 Arithmetic progressions
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##### References:
 [1] Alon, N.; Asodi, V., Tracing a single user, European J. Combin., 27, 1227-1234, (2006) · Zbl 1105.05068 [2] Alon, N.; Asodi, V., Tracing many users with almost no rate penalty, IEEE Trans. Inform. Theory, 53, 437-439, (2007) · Zbl 1310.94230 [3] Alon, N.; Shapira, A., On an extremal hypergraph problem of Brown, Erdős and Sós, Combinatorica, 26, 627-645, (2006) · Zbl 1121.05079 [4] Babai, L.; Sós, V. T., Sidon sets in groups and induced subgroups of Cayley graphs, European J. Combin., 6, 101-114, (1985) · Zbl 0573.05032 [5] Baranyai, Zs., On the factorization of the complete uniform hypergraph, (Infinite and Finite Sets, Keszthely, Hungary, 1973. Vol. I, Proc. Colloq. Math. Soc. János Bolyai, vol. 10, (1975), North-Holland Amsterdam), 91-108 · Zbl 0306.05137 [6] Behrend, F. A., On sets of integers which contain no three terms in arihtmetical progression, Proc. Nat. Acad. Sci. USA, 32, 331-332, (1946) · Zbl 0060.10302 [7] A. Blokhuis, Private communication, June 16, 2009. [8] A.E. Brouwer, Steiner Triple Systems Without Forbidden Subconfigurations, Mathematisch Centrum Amsterdam, ZW 104/77, 1977. · Zbl 0367.05011 [9] Brown, W. G., On graphs that do not contain a thomsen graph, Canad. Math. Bull., 9, 281-289, (1966) · Zbl 0178.27302 [10] Brown, W. G.; Erdős, P.; Sós, V. T., On the existence of triangulated spheres in 3-graphs and related problems, Period. Math. Hungar., 3, 221-228, (1973) · Zbl 0269.05111 [11] Brown, W. G.; Erdős, P.; Sós, V. T., Some extremal problems on $$r$$-graphs, (Proceedings of the Third Annual Arbor Conference on Graph Theory, New Directions in the Theory of the Graphs, (1973), Academic Press New York), 55-63 [12] Caro, Y.; Yuster, R., Packing graphs: the packing problem solved, Electron. J. Combin., 4, 7, (1997), Research Paper 1 · Zbl 0885.05052 [13] Colbourn, C. J.; Mendelsohn, E.; Rosa, A.; Širáň, J., Anti-mitre Steiner triple systems, Graphs Combin., 10, 215-224, (1994) · Zbl 0815.05017 [14] Colbourn, C. J.; Rosa, A., Triple systems, (1999), Clarendon Press, Oxford University Press New York · Zbl 0938.05009 [15] C.J. Colbourn, Private communication, Nov. 15, 2004. [16] De Bonis, A.; Vaccaro, U., Optimal algorithms for two group testing problems and new bounds on generalized superimposed codes, IEEE Trans. Inform. Theory, 52, 4673-4680, (2006) · Zbl 1320.94056 [17] D’yachkov, A. G.; Rykov, V. V., Bounds on the length of disjunctive codes, Problemy Peredaci Informacii, 18, 7-13, (1982) · Zbl 0507.94013 [18] D’yachkov, A. G.; Rykov, V. V., Optimal superimposed codes and designs for Rényi’s search model, J. Statist. Plann. Inference, 100, 281-302, (2002) · Zbl 0998.94043 [19] Erdős, P., Extremal problems in graph theory, (Fiedler, M., Theory of Graphs and its Applications, (1964), Academic Press New York), 29-36 [20] Erdős, P., On extremal problems of graphs and generalized graphs, Israel J. Math., 2, 183-190, (1964) · Zbl 0129.39905 [21] Erdős, P., Problems and results in combinatorial analysis, (Colloq. Internazionale Teorie Combinatorie (Rome, 1973), Tomo II, Atti dei Convegni Lincei, vol. 17, (1976), Accad. Naz. Lincei Rome), 3-17 [22] Erdős, P.; Frankl, P.; Füredi, Z., Families of finite sets in which no set is covered by the union of two others, J. Combin. Theory Ser. A, 33, 158-166, (1982) · Zbl 0489.05003 [23] Erdős, P.; Frankl, P.; Füredi, Z., Families of finite sets in which no set is covered by the union of $$r$$ others, Israel J. Math., 51, 79-89, (1985) · Zbl 0587.05021 [24] Erdős, P.; Frankl, P.; Rödl, V., The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin., 2, 113-121, (1986) · Zbl 0593.05038 [25] Erdős, P.; Rényi, A.; Sós, V. T., On a problem of graph theory, Studia Sci. Math. Hungar., 1, 215-235, (1966) · Zbl 0144.23302 [26] Erdős, P.; Simonovits, M., Compactness results in extremal graph theory, Combinatorica, 2, 275-288, (1982) · Zbl 0508.05043 [27] Erdős, P.; Turán, P., On a problem of sidon in additive number theory and some related problems, J. Lond. Math. Soc., 16, 212-215, (1941) · JFM 67.0984.03 [28] Ericson, T.; Györfi, L., Superimposed codes in $$R^n$$, IEEE Trans. Inform. Theory, 34, 877-880, (1988) · Zbl 0657.94014 [29] Forbes, A. D.; Grannell, M. J.; Griggs, T. S., On 6-sparse Steiner triple systems, J. Combin. Theory Ser. A, 144, 235-252, (2007) · Zbl 1121.05015 [30] Forbes, A. D.; Grannell, M. J.; Griggs, T. S., Further 6-sparse Steiner triple systems, Graphs Combin., 25, 49-64, (2009) · Zbl 1188.05034 [31] Frankl, P.; Füredi, Z., A new extremal property of Steiner triple systems, Discrete Math., 48, 205-212, (1984) · Zbl 0553.05019 [32] Frankl, P.; Füredi, Z., Union-free families of sets and equations over fields, J. Number Theory, 23, 210-218, (1986) · Zbl 0589.05013 [33] Frankl, P.; Füredi, Z., Exact solution of some Turán-type problems, J. Combin. Theory, Ser. A, 45, 226-262, (1987) · Zbl 0661.05003 [34] Fujiwara, Y., Infinite classes of anti-mitre and 5-sparse Steiner triple systems, J. Combin. Des., 14, 237-250, (2006) · Zbl 1089.05012 [35] Füredi, Z., A note on $$r$$-cover-free families, J. Combin. Theory Ser. A, 73, 172-173, (1996) · Zbl 0843.05100 [36] Z. Füredi, M. Ruszinkó, Superimposed codes are almost big distance ones, in: Proc. 1997 IEEE Int. Symp. Inform. Theory, Ulm, Germany, June 29-July 4, 1997, p. 118. [37] Füredi, Z.; Ruszinkó, M., An improved upper bound of the rate of Euclidean superimposed codes, IEEE Trans. Inform. Theory, 45, 799-802, (1999) · Zbl 0945.94031 [38] Grannell, M. J.; Griggs, T. S.; Whitehead, C. A., The resolution of the anti-pasch conjecture, J. Combin. Des., 8, 300-309, (2000) · Zbl 0959.05016 [39] Griggs, T. S.; Murphy, J. P., 101 anti-pasch Steiner triple systems of order 19, J. Combin. Math. Combin. Comput., 13, 129-141, (1993) · Zbl 0778.05022 [40] Griggs, T. S.; Murphy, J.; Phelan, J. S., Anti-pasch Steiner triple systems, J. Combin. Inf. Syst. Sci., 15, 79-84, (1990) · Zbl 0741.05009 [41] Heath-Brown, D. R., Integer sets containing no arithmetic progression, J. Lond. Math. Soc., 35, 385-394, (1987) · Zbl 0589.10062 [42] Hwang, F. K., A method for detecting all defective members in a population by group testing, J. Amer. Statist. Assoc., 67, 339, 605-608, (1972) · Zbl 0247.62010 [43] Hwang, F. K.; Sós, V. T., Non adaptive hypergeometric group testing, Studia Sci. Math. Hungar., 22, 257-263, (1987) · Zbl 0639.62076 [44] Iwaniec, H.; Pintz, J., Primes in short intervals, Monatsh. Math., 98, 115-143, (1984) · Zbl 0544.10040 [45] Johnson, S. M., On the upper bounds for unrestricted binary error-correcting codes, IEEE Trans. Inform. Theory, 17, 466-478, (1971) · Zbl 0231.94008 [46] Johnson, S. M., A new upper bound for error-correcting codes, IRE Trans. Inform. Theory, 8, 203-207, (1962) · Zbl 0102.34602 [47] Johnson, S. M., Improved asymptotic bounds for error-correcting codes, IEEE Trans. Inform. Theory, 9, 198-205, (1963) · Zbl 0282.94010 [48] Kautz, W. H.; Singleton, R. C., Nonrandom binary superimposed codes, IEEE Trans. Inform. Theory, 10, 363-377, (1964) · Zbl 0133.12402 [49] Kim, H. K.; Lebedev, V., On optimal superimposed codes, J. Combin. Des., 12, 79-91, (2004) · Zbl 1051.94013 [50] Kim, H. K.; Lebedev, V.; Oh, D. Y., Some new results on superimposed codes, J. Combin. Des., 13, 276-285, (2005) · Zbl 1169.94350 [51] Ling, A. C.H., A direct product construction for 5-sparse triple systems, J. Combin. Des., 5, 443-447, (1997) · Zbl 0912.05012 [52] Ling, A. C.H.; Colbourn, C. J.; Grannell, M. J.; Griggs, T. S., Construction techniques for anti-pasch Steiner triple systems, J. Lond. Math. Soc. (2), 61, 641-657, (2000) · Zbl 0956.05023 [53] Macula, A. J., A simple construction of $$d$$-disjunct matrices with certain constant weights, Discrete Math., 162, 311-312, (1996) · Zbl 0870.05012 [54] H. Minkowski, Geometrie und Zahlen, Leipzig und Berlin, 1896. [55] Nguyen Quang, A.; Zeisel, T., Bounds on constant weight binary superimposed codes, Probl. Control Inf. Theory, 17, 223-230, (1988) · Zbl 0652.94021 [56] Ray-Chaudhuri, D. K.; Wilson, R. M., The existence of resolvable block designs, (Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), (1973), North-Holland Amsterdam), 361-375 [57] Ruszinkó, M., On the upper bound of the size of the $$r$$-cover-free families, J. Combin. Theory Ser. A, 66, 302-310, (1994) · Zbl 0798.05071 [58] Ruzsa, I. Z., Solving a linear equation in a set of integers I, Acta Arith., 65, 259-282, (1993) · Zbl 1042.11525 [59] I.Z. Ruzsa, E. Szemerédi, Triple systems with no six points carrying three triangles, in: Combinatorics, Keszthely, 1976, Coll. Math. Soc. J. Bolyai 18 939-945. Volume II. [60] Sárközy, G. N.; Selkow, S., An extension of the ruzsa-Szemerédi theorem, Combinatorica, 25, 77-84, (2005) · Zbl 1065.05068 [61] Sárközy, G. N.; Selkow, S., On a Turán-type hypergraph problem of Brown, Erdős and T. Sós, Discrete Math., 297, 190-195, (2005) · Zbl 1079.05063 [62] Shapira, A., Behrend-type constructions for sets of linear equations, Acta Arith., 122, 17-33, (2006) · Zbl 1149.11012 [63] Sós, V. T., An additive problem in different structures, (Proc. of the Second Int. Conf. in Graph Theory, Combinatorics, Algorithms, and Applications, San Fra. Univ., California, July 1989, (1991), SIAM Philadelphia), 486-510 · Zbl 0760.05088 [64] Szemerédi, E., On sets of integers containing no $$k$$ elements in arithmetic progression, Acta Arith., 27, 199-245, (1975) · Zbl 0303.10056 [65] Szemerédi, E., Integer sets containing no arithmetic progression, Acta Math. Hungar., 56, 155-158, (1990) · Zbl 0721.11007 [66] L. Teirlinck, Review MR2455590 (2009i:05037) on [70]. Mathematical Reviews, 2009. [67] Wilson, R. M., On existence theory for pairwise balanced designs, I, II, III, J. Combin. Theory Ser. A, 13, 220-245, (1972), 246-273, 18 (1975), 71-79 · Zbl 0263.05014 [68] Wolfe, A., 5-sparse Steiner triple systems of order $$n$$ exist for almost all admissible $$n$$, Electron. J. Combin., 12, 42, (2005), Research Paper 68 · Zbl 1079.05013 [69] Wolfe, A., The resolution of the anti-mitre Steiner triple system conjecture, J. Combin. Des., 14, 229-236, (2006) · Zbl 1089.05013 [70] Wolfe, A. J., The existence of 5-sparse Steiner triple systems of order $$n \equiv 3 \sim(\operatorname{mod} 6)$$, $$n \notin \{9, 15 \}$$, J. Combin. Theory Ser. A, 115, 1487-1503, (2008) · Zbl 1157.05011
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