On the densities of cliques and independent sets in graphs

Abstract

Let r, s ≥ 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph Kn is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.

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

“…At this writing even 3 is not yet fully understood (but see [11,16]). The state of our knowledge of l for l ≥ 4 is really very limited, though some work already exists, e.g., [7,8,12,13,[17][18][19]. Much of the recent progress in this area was achieved using Razborov's flag algebras method.…”

On the 3‐Local Profiles of Graphs

“…On the other hand, the max-min version of this problem is now solved. As we have recently proved [14], max G min{d(K r ; G), d(K r ; G)} is obtained by a clique on a properly chosen fraction of the vertices.…”

“…A simple consequence of Goodman's inequality is that min G max{d(K 3 ; G), d(K 3 ; G)} = 1/8. The analogous statement for r = 4 is not true as can be shown using an example of Franek and Rödl [8] (see [14] for the details). On the other hand, the max-min version of this problem is now solved.…”

Permutations with fixed pattern densities

“…This multiplicity of solutions, if there are no permutons bridging the gap, suggests a phase transition in the entropy‐optimal permuton in the interior of in a neighborhood of the dimple. In fact, we can use a stability theorem from to show that the phenomenon is real.…”

“…Theorem 17 (special case of Theorems 1.1 and 1.2 of [18]). For any > 0 there is a > 0 and an N such that for any n-vertex graph G with n > N and (K 3 , G) ≥ p and | (K 3 , G) − M p | < is -close to a graph H on n vertices consisting of a clique and isolated vertices, or the complement of a graph consisting of a clique and isolated vertices.…”