TY - JOUR

T1 - A problem of Erdos on the minimum number of k-cliques

AU - Das, Shagnik

AU - Huang, Hao

AU - Ma, Jie

AU - Naves, Humberto

AU - Sudakov, Benny

PY - 2013/5/1

Y1 - 2013/5/1

N2 - Fifty years ago Erdos asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l-1 complete graphs of size nl-1. This conjecture was disproved by Nikiforov, who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2.In this paper we solve Erdos' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdos' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).

AB - Fifty years ago Erdos asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l-1 complete graphs of size nl-1. This conjecture was disproved by Nikiforov, who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2.In this paper we solve Erdos' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdos' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).

KW - Clique density

KW - Erdos' conjecture

KW - Flag algebras

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U2 - 10.1016/j.jctb.2013.02.003

DO - 10.1016/j.jctb.2013.02.003

M3 - Article

AN - SCOPUS:84877136166

VL - 103

SP - 344

EP - 373

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 3

ER -