On problems of Erdős and Baumann-Briggs on minimising the density of $s$-cliques in graphs with forbidden subgraphs

Abstract: 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 $2<sup>{1-s}+o(1)$ 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.

• The paper solves the Erdős problem for (s,ℓ)=(8,3), proving er(8,\overline{K_3})=491411/268435456 and showing extremal graphs are expansions of the Clebsch-graph complement R_{3,3,3}.
• The authors use flag algebras, high-precision semidefinite certificates, and perfect-stability methods to obtain both asymptotic closeness and exact extremal structure for sufficiently large graphs.
• The paper develops a reduction from Baumann–Briggs constraints to Erdős-type forbidden-subgraph problems, yielding asymptotic results for several (k,t,s) ranges while identifying cases that require different methods.

Overview

The paper by Bodnár and Pikhurko (2602.17412) studies two related extremal problems on minimizing the density of $s$-cliques in large graphs subject to induced-subgraph constraints. The first is a problem of Erdős from 1962: determine $ER(s,n,\overline{K_\ell})$, the minimum number of $s$-cliques in an $n$-vertex graph with no independent set of size $\ell$. The second, posed recently by Baumann and Briggs, asks for $ER(s,n,\mathcal{B}_{k,t})$, the minimum number of $s$-cliques in an $n$-vertex graph in which every $k$-vertex set spans a $t$-clique. The paper's main contribution is a new solved case of Erdős's problem — the case $ER(s,n,\overline{K_\ell})$0 — together with a general reduction principle that translates Erdős-type results into asymptotic answers for the Baumann–Briggs problem.

The new Erdős case: $ER(s,n,\overline{K_\ell})$1, $ER(s,n,\overline{K_\ell})$2

The central result is that every almost $ER(s,n,\overline{K_\ell})$3-extremal graph is $ER(s,n,\overline{K_\ell})$4-close in edit distance to a uniform expansion of $ER(s,n,\overline{K_\ell})$5, the complement of the Clebsch graph (a 16-vertex, 10-regular, $ER(s,n,\overline{K_\ell})$6-free graph with independence number 3), and that for all sufficiently large $ER(s,n,\overline{K_\ell})$7 every extremal graph is exactly an expansion of $ER(s,n,\overline{K_\ell})$8. Consequently,

$ER(s,n,\overline{K_\ell})$9

resolving a new case of Erdős's problem. The value follows from the general clique-density formula for uniform expansions of $s$0, namely $s$1 evaluated at $s$2.

The proof uses Razborov's flag algebra method with base flags of 8 vertices, at which point there are 410 triangle-free graphs up to isomorphism. The certificates were produced with the authors' FlagAlgebraToolbox SageMath package and solved with the high-precision SDP solver SDPA-QD; the Jupyter notebook and certificates are included as ancillary files. Crucially, the certificate also satisfies the sufficient condition of Pikhurko, Sliačan and Tyros for perfect $s$3-stability, which yields the full stability and exact-structure conclusions rather than merely the density value. The verification is computer-assisted; some supporting structural facts (e.g., the uniqueness of embedding $s$4 into an expansion of $s$5) were established by hand.

Further Erdős-type results with additional forbidden subgraphs

Two theorems of independent interest forbid a clique plus one extra graph:

• For $s$6, forbidding $s$7 and the induced 5-cycle $s$8 gives $s$9, with stability towards uniform expansions of $n$0 (two cliques), and exact structure: for large $n$1, every extremal graph contains a uniform expansion of $n$2 as a spanning subgraph and $n$3.
• For $n$4, forbidding $n$5 and $n$6 (the 5-cycle plus an isolated vertex) gives $n$7, with stability towards uniform expansions of $n$8 and the analogous exact extremal count.

The stability proofs are non-computer-based: the flag algebra certificates show that the density of $n$9 (a clique with one edge missing) is $\ell$0, and then a Ramsey-type argument extracts a large clique whose complement structure forces the graph to be a union of $\ell$1 cliques plus $\ell$2 cross-edges. Convexity then forces the parts to be balanced. The exact extremal analysis proceeds by maximizing internal edges over partitions, showing that the set of "wrong pairs" has maximum degree $\ell$3, and deriving contradictions from any deviation from the expansion structure.

Reduction from Baumann–Briggs to Erdős-type problems

The bridge between the two problems is Proposition (Reduction): if $\ell$4 (graphs admitting a $\ell$5-vertex expansion with no $\ell$6-clique) and $\ell$7 is such that expansions of $\ell$8 are almost extremal for $\ell$9, then $ER(s,n,\mathcal{B}_{k,t})$0, and almost extremal graphs for the Baumann–Briggs problem are $ER(s,n,\mathcal{B}_{k,t})$1-close to almost extremal Erdős graphs. The proof uses the Induced Removal Lemma of Alon, Fischer, Krivelevich and Szegedy together with Erdős's $ER(s,n,\mathcal{B}_{k,t})$2-partite hypergraph matching result to convert a $ER(s,n,\mathcal{B}_{k,t})$3-free graph into a $ER(s,n,\mathcal{B}_{k,t})$4-free one with $ER(s,n,\mathcal{B}_{k,t})$5 edits.

Applying this requires computing, for each candidate host graph $ER(s,n,\mathcal{B}_{k,t})$6, the function $ER(s,n,\mathcal{B}_{k,t})$7 — the largest $ER(s,n,\mathcal{B}_{k,t})$8 such that some $ER(s,n,\mathcal{B}_{k,t})$9-vertex expansion of $s$0 is $s$1-free. The paper computes these exactly:

• $s$2 (a pigeonhole argument).
• $s$3, via a counting lemma (each vertex lies in exactly $s$4 $s$5-cliques forces $s$6 when every vertex of $s$7 lies in exactly $s$8 $s$9-cliques) plus explicit constructions.
• $n$0 and $n$1, with the even case requiring a delicate matching-based argument.
• $n$2 for $n$3, with the $n$4 cases requiring non-uniform part sizes (including one part smaller than $n$5).

Consequences for the Baumann–Briggs problem

The reduction yields asymptotic solutions for numerous parameter ranges with $n$6 and $n$7:

$n$8	$n$9	$k$0	Extremal host
$k$1	$k$2	$k$3	$k$4
$k$5	$k$6	$k$7	$k$8
$k$9	$t$0	$t$1	$t$2
$t$3	$t$4	expression above	$t$5
$t$6	$t$7	$t$8	$t$9
$ER(s,n,\overline{K_\ell})$00	$ER(s,n,\overline{K_\ell})$01	$ER(s,n,\overline{K_\ell})$02	$ER(s,n,\overline{K_\ell})$03

Additionally, $ER(s,n,\overline{K_\ell})$04 whenever $ER(s,n,\overline{K_\ell})$05, by a Kővári–Sós–Turán argument. The case $ER(s,n,\overline{K_\ell})$06 answers asymptotically Problem 19 of Baumann and Briggs, and a companion theorem gives the exact extremal structure for $ER(s,n,\overline{K_\ell})$07: for $ER(s,n,\overline{K_\ell})$08 and large $ER(s,n,\overline{K_\ell})$09, every extremal graph is a uniform expansion of $ER(s,n,\overline{K_\ell})$10 with exactly $ER(s,n,\overline{K_\ell})$11 cliques.

Notably, the reduction is not universal: the authors show by computer search that no pair $ER(s,n,\overline{K_\ell})$12 with $ER(s,n,\overline{K_\ell})$13 allows $ER(s,n,\overline{K_\ell})$14 to serve as the extremal host via this route, and that for $ER(s,n,\overline{K_\ell})$15 the value is strictly below $ER(s,n,\overline{K_\ell})$16 because non-uniform expansions of the 8-vertex $ER(s,n,\overline{K_\ell})$17-Ramsey graph $ER(s,n,\overline{K_\ell})$18 achieve $ER(s,n,\overline{K_\ell})$19. Thus some Baumann–Briggs cases do not reduce to any instance of the Erdős problem.

Limitations and open questions

The paper concedes that exact (non-asymptotic) values of the Baumann–Briggs function are not determined except when $ER(s,n,\overline{K_\ell})$20; the authors state that handling cases such as $ER(s,n,\overline{K_\ell})$21, where edges (e.g., a star) may be deleted from parts of an expansion without creating forbidden subgraphs, would involve "technical work rather than new insights" and is not pursued. The restriction $ER(s,n,\overline{K_\ell})$22 in the $ER(s,n,\overline{K_\ell})$23 theorem is conjectured to be unnecessary — the authors conjecture $ER(s,n,\overline{K_\ell})$24 for all $ER(s,n,\overline{K_\ell})$25 — but this remains open. The main density results are computer-assisted: they rest on floating-point semidefinite certificates (verified with SDPA-QD at high precision) rather than fully hand-checkable proofs, and the stability conclusions depend on the perfect-stability criterion of Pikhurko–Sliačan–Tyros being satisfiable by the computed certificate.

Conclusion

The paper resolves the $ER(s,n,\overline{K_\ell})$26 case of Erdős's 1962 clique-minimization problem with the exact value $ER(s,n,\overline{K_\ell})$27, establishes stability and exact extremal structure for that and several related forbidden-subgraph problems, and provides a general reduction principle that converts such results into asymptotic solutions of the Baumann–Briggs problem across a range of parameters. The work leaves open the extension to $ER(s,n,\overline{K_\ell})$28 in the $ER(s,n,\overline{K_\ell})$29 case, exact values for general $ER(s,n,\overline{K_\ell})$30, and a characterization of Baumann–Briggs instances that cannot be captured by the reduction to Erdős-type problems.