On Arithmetic Progressions of Cycle Lengths in Graphs
Combinatorics, Probability and Computing
Batch codes and their applications
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
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Let n,r,k be positive integers such that 3@?k~. This problem is related to the forbidden hypergraph problem introduced by Brown, Erdos, and Sos and very recently discussed in the context of combinatorial batch codes (CBCs). In this short paper we obtain the following results. (i)Using a result due to Erdos we are able to show m(n,r,k)=o(n^r) for 7@?k, and 3@?r@?k-1-@?logk@?. This result is best possible with respect to the upper bound on r as we subsequently show through explicit construction that for 6@?k, and k-@?logk@?@?r@?k-1,m(n,r,k)=@Q(n^r). This explicit construction improves on the non-constructive general lower bound obtained by Brown, Erdos, and Sos for the considered parameter values. (ii)For 2-uniform CBCs we obtain the following results. (a)We provide exact value of m(n,2,5) for n=5. (b)Using a result of Lazebnik et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n,2,k) (@W(n^k^+^1^k^-^1)) for all k=8 and infinitely many values of n. (c)We show m(n,2,k)=O(n^1^+^1^@?^k^4^@?) by using a result due to Bondy and Simonovits, and also show m(n,2,k)=@Q(n^3^2) for k=6,7,8 by using a result of Kovari, Sos, and Turan.