A problem of Erdős on the minimum number of k -cliques

Abstract

Fifty years ago Erdős 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 n l−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 n 2 . In this paper we solve Erdős' problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdős' conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).

Quasi‐carousel tournaments

“…Using the flag algebra semidefinite method, we were able to obtain the bound subject to floating point rounding errors. This suggests that the conjecture is true and that there may be a straightforward (but numerically intensive) proof using the semidefinite method and rounding techniques (see for some examples).…”

“…Using the flag algebra semidefinite method, we were able to obtain the bound ∀ ∈ Hom + ( 0 , ℝ), ( 4 ) ⩽ 0.157516, subject to floating point rounding errors. This suggests that the conjecture is true and that there may be a straightforward (but numerically intensive) proof using the semidefinite method and rounding techniques (see [1,10,11,14,22] for some examples). The intuition of the recursive construction of , is that at every step we have one part −1 ⧵ that maximizes the density of ⃗ 3 (hence is almost balanced) and another part whose vertices all beat the first part.…”

On the 3‐Local Profiles of Graphs

“…1 2 ]×[0,1] is a continuous map from a topological 2-disc, mapping its boundary to a path encircling C , 1]. Therefore, as before, the region between these curves is contained in (K 3…”

“…The case α = 0 of this problem was posed by Erdős more than 50 years ago. Although, recently Das et al [3], and independently Pikhurko [18], solved it for certain values of r and s it is still open in general.…”

On the densities of cliques and independent sets in graphs

“…We showed that if the density of independent sets of size r is fixed, the maximum density of s-cliques is achieved when the graph itself is either a clique on a subset of the vertices, or a complement of a clique. On the other hand, the problem of minimizing the clique density seems much harder and has quite different extremal graphs for various values of r and s (at least when α = 0, see [3,13]). Question 7.1.…”