Using flag algebras, we prove that the minimum density of $8$-cliques in a large graph without an independent set of size $3$ is $491411/268435456+o(1)$, thus resolving a new case of an old problem of Erdős [Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962) 459-464]. Also, we establish some other results of this type; for example, we show that the minimum $s$-clique density in a large graph with no independent set of size 3 nor an induced 5-cycle is $2^{1-s}+o(1)$ when $s=4,5,6$. For each of these results, we also describe the structure of all extremal and almost extremal graphs of large order $n$. These results are applied to give an asymptotic solution to a number of cases of the problem of Baumann and Briggs [Electronic J Comb 32 (2025) P1.22] which asks for the minimum number of $s$-cliques in an $n$-vertex graph in which every $k$-set spans a $t$-clique.
As usual, a graph G is a pair (V (G), E(G)), where V (G) is the vertex set and the edge set E(G) consists of unordered pairs of vertices. We let G := (V (G), V (G) 2 \ E(G)) denote its complement and v(G) := |V (G)| denote its order. For graphs F and G, let P (F, G) be the number of v(F )-subsets of V (G) that induce in G a subgraph isomorphic to F . Let K ℓ denote the ℓ-clique, that is, the complete graph with ℓ vertices. Thus, for example, K ℓ is the empty graph on ℓ vertices.
Given a family F of graphs and an integer s, we consider the following extremal function: ER s (n, F) := min{P (K s , G) : v(G) = n, ∀ F ∈ F P (F, G) = 0}, for n ∈ N, (1) the minimum number of s-cliques in an n-vertex graph G which is F -free (that is, no F ∈ F can appear in G as an induced subgraph). If the graph family F = {F } consists of a single graph, then we write ER s (n, F ) for ER s (n, {F }), etc.
The case s = 2 of (1) amounts, after passing to graph complements, to the classical Turán function. This was fully resolved by Turán [27] when any given clique is forbidden (see also Mantel [17]), while the Erdős-Stone Theorem [10] determines it within an additive o(n 2 ) error term for any forbidden family (with the corresponding stability result proved by Erdős [9] and Simonovits [26]). So we are interested in the case s ⩾ 3.
The special case of (1) when s = 3 and F = {K 3 } was resolved by Goodman [12] (for all even n), with Lorden [16] determining the exact value for all odd n. Erdős [7] asked for the value of ER s (n, K ℓ ) and made a too optimistic conjecture about it which was disproved by Nikiforov [19]. It is easy to show that the limit er s (F) := lim n→∞ ER s (n, F) n s (2) exists for every graph family F (see e.g. [22,Lemma 2.2] for a proof). However, the value of er s (K ℓ ) for any other non-trivial pair (s, ℓ) apart from those mentioned above was not known for a long while.
After the introduction of the flag algebra method by Razborov [24,25], a number of new cases have been solved. In order to state these results, we need some further definitions. For an integer m ⩾ 0, we denote [m] := {0, . . . , m -1}. Let C m denote the m-cycle whose vertex set is [m] with the edge set consisting of the pairs of consecutive vertices modulo m. Let R 3,5 denote the graph on [13] = {0, . . . , 12} in which x, y ∈ V (R) are adjacent if x -y modulo 13 is in {1, 5, 8, 12}. It is easy to check that R 3,5 does not contain K 3 nor K 5 as induced subgraphs. (In fact, R 3,5 is the unique (3,5)-Ramsey graph on 13 vertices.) Let R 3,3,3 be the graph whose vertex set consists of all even-sized subsets of [5], where
that is, their symmetric difference has 2 elements. Thus R 3,3,3 is a 10-regular graph with 16 vertices. It does not contain K 6 nor K 3 . In fact, R 3,3,3 is the complement of the Clebsch graph (which appears in extremal (3, 3, 3)-Ramsey colourings). Given a graph H, which by relabelling its vertices is assumed to have vertex set [m], and disjoint sets V 0 , . . . , V m-1 , let the expansion H((V 0 , . . . , V m-1 )) be the graph on V 0 ∪ • • • ∪ V m-1 obtained by putting the complete graph on each V i and putting, for each edge {i, j} ∈ E(H), the complete bipartite graph between V i and V j . An expansion is uniform if
If we consider expansion in terms of complements, then it amounts to blowing up each vertex i of H by factor |V i |. The edit distance between two graphs of the same order is the smallest number of adjacencies that have to be changed in one graph to make it isomorphic to the other. We call a graph ER s (n, F)-extremal if it has n vertices, is F-free and has ER s (n, F) s-cliques. We call a graph G of order n (or rather a sequence of graphs with
If s, F and n are understood, we may just say (almost) extremal. Now we are ready to state the following theorem that collects some of the results established by Das, Huang, Ma, Naves, and Sudakov [5], Parczyk, Pokutta, Spiegel and Szabó [21], and Pikhurko and Vaughan [23] (in addition to the results from [12,16] for which the stated stability property can be easily established by modern methods).
Theorem 1 ([5, 12, 16, 21, 23]) Let (s, ℓ) be one of the pairs of integers listed below and let H be the following graph:
Then every almost ER s (n, K ℓ )-extremal graph is o(n 2 )-close in the edit distance to a uniform expansion of H. Moreover, for all sufficiently large n, every ER s (n, K ℓ )-extremal graph contains an expansion of H as a spanning subgraph.
Formally, the first conclusion of Theorem 1 states that for every ε > 0 there are δ > 0 and n 0 such that every K ℓ -free graph G with n ⩾ n 0 vertices and
2 from a uniform expansion of H. When the meaning is clear, we will be using the shorter o(1)-version and refer to this property (that all almost extremal n-vertex graphs are within edit distance o(n 2 ) from each other) as stability. In Cases (b)-(d), it holds that, in fact, every large ER s (n, K ℓ )-extremal graph is an expansion of H (without any extra ed
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