Minimum Number ofk-Cliques in Graphs with Bounded Independence Number

Abstract

Erdős asked in 1962 about the value of f (n, k, l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k, l) = (3, 3) was solved by Lorden. Here we solve the problem (for all large n) for (3, l) with 4 ≤ l ≤ 7 and (k, 3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves, and Sudakov resolved the cases (k, l) = (3, 4) and (4, 3).

Graphs with Few 3‐Cliques and 3‐Anticliques are 3‐Universal

“…We note that it is an old and intriguing problem how small p(K k , G) can be for a large k-clique-free graph. See [4] and the recent work [14]. (8) What are the possible values of t k (T ), given the value of t l (T )?…”

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 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.…”

“…This is a fifty-year-old question of Erdős, which is still widely open. Das et al [3], and independently Pikhurko [13], solved this problem for certain values of r and s. It would be interesting if one could describe how the extremal graph changes as α goes from 0 to 1 in these cases. As mentioned in the introduction, the problem of minimizing d(K s ; G) in graphs with fixed density of r-cliques for r < s is also open and so far solved only when r = 2.…”