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arxiv: 2307.06455 · v4 · submitted 2023-07-12 · 🧮 math.CO

Induced subgraph density. IV. New graphs with the ErdH{o}s-Hajnal property

Tung Nguyen , Alex Scott , Paul Seymour This is my paper

Pith reviewed 2026-05-24 08:02 UTC · model grok-4.3

classification 🧮 math.CO
keywords Erdős-Hajnal conjectureinduced subgraphsprime graphsbuildable graphsiterative sparsificationcliquesstable setstournaments
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The pith

Graphs where every prime induced subgraph has a degree-one vertex satisfy the Erdős-Hajnal conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors establish that any graph H with the property that every prime induced subgraph of size at least three has a vertex of degree one satisfies the Erdős-Hajnal conjecture. This property, called buildable, allows them to prove that forbidding such an H1 and the complement of another buildable H2 guarantees a clique or stable set of size polynomial in the graph size. They show there are infinitely many prime graphs with this property. The proof introduces iterative sparsification to repeatedly restrict the graph while preserving the bound. This also yields new results for ordered graphs and tournaments.

Core claim

If H1 and the complement of H2 are buildable, meaning every prime induced subgraph with at least three vertices has a vertex of degree one, then every graph G free of both H1 and H2 has a clique or stable set of size at least |G|^c for some positive c. This holds in particular for any single buildable H, including infinitely many primes that also have a vertex of degree |G'|-2 in each prime subgraph.

What carries the argument

Buildable graphs, where every prime induced subgraph of size at least three has a vertex of degree one, together with the iterative sparsification technique that passes to a sequence of successively more restricted induced subgraphs.

If this is right

  • Every buildable graph H satisfies the Erdős-Hajnal conjecture.
  • Infinitely many prime graphs satisfy the Erdős-Hajnal conjecture.
  • If H1 and the complement of H2 are buildable, every (H1, H2)-free graph has large clique or stable set.
  • The result extends to ordered graphs, extending previous bounds on monotone paths.
  • Infinitely many new prime tournaments satisfy the Erdős-Hajnal conjecture in tournament form.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may help resolve the conjecture for additional families of graphs beyond buildable ones.
  • The iterative sparsification could be adapted to other problems involving induced subgraphs or forbidden patterns.
  • Connections between graph, ordered graph, and tournament versions suggest unified methods for extremal problems in these areas.

Load-bearing premise

The iterative sparsification process can be repeated indefinitely while keeping the graphs buildable and maintaining a polynomial bound on the size of the largest clique or stable set.

What would settle it

A counterexample would be a buildable graph H together with an H-free graph whose largest clique and stable set are both smaller than any positive power of the total number of vertices.

Figures

Figures reproduced from arXiv: 2307.06455 by Alex Scott, Paul Seymour, Tung Nguyen.

Figure 1
Figure 1. Figure 1: The two six-vertex prime graphs in H, and one on seven vertices. We define H to be the class of all graphs that can be constructed by a sequence of the following operations, starting with one-vertex graphs: • choosing a graph G that is already constructed, choosing a vertex v ∈ V (G) that has degree at least |G| − 1, and adding a new vertex adjacent only to v; • choosing a graph G that is already construct… view at source ↗
Figure 2
Figure 2. Figure 2: Start with a path (a2-b3-a6-b7-a10-b11 in this case), add a leaf at every vertex, add an isolated vertex b1, and take a bipartition (A, B), numbered as shown. Now make A a clique; and make ai , bj adjacent if i ≥ j + 4. • each prime graph in H is a split graph, that is, its vertex set is the union of a clique and a stable set; and • H ∈ H if and only if every nontrivial prime induced subgraph H′ has a vert… view at source ↗
Figure 3
Figure 3. Figure 3: The six-vertex graphs not containing P5 or P5 that remain open. We will actually prove a statement stronger than Theorem 1.4 (in fact, our proof method requires most of this additional strength). Our final result is stronger than Theorem 1.4 in four ways: • We will prove that these graphs satisfy a conjecture of Fox and Sudakov (explained below), not just that they have the Erd˝os-Hajnal property. • We can… view at source ↗
Figure 4
Figure 4. Figure 4: With H as shown, H ∈ J , and so {H, H} is viral. We define J to be the class of all graphs that can be constructed by a sequence of the following operations, starting with one-vertex graphs: • choosing a graph G that is already constructed, choosing a vertex v ∈ V (G), and adding a new vertex adjacent only to v; • choosing two graphs H1, H2 that are already constructed, and substituting H2 for a vertex of … view at source ↗
Figure 5
Figure 5. Figure 5: The four-vertex prime ordered graphs (up to complements and reversal). Next, let us look at corollaries of Theorem 2.1 for unordered graphs. We will use a nice correspondence between the classes L, K of ordered graphs and the classes H,J of (unordered) graphs: Lemma 7.3. If G ∈ K then G♮ ∈ J ; and if F ∈ J , there is a linear order ≤ of V (F) such that (F, ≤) ∈ K. Similarly, if G ∈ L then G♮ ∈ H; and if F … view at source ↗
read the original abstract

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$). Alon, Pach, and Solymosi reduced the Erd\H{o}s-Hajnal conjecture to the case when $H$ is {\em prime} (that is, $H$ cannot be obtained by vertex-substitution from smaller graphs); but until now, it was not shown for any prime graph with more than five vertices. We will provide infinitely many prime graphs that satisfy the conjecture. Let $H$ be a graph with the property that for every prime induced subgraph $G'$ with $|G'|\ge 3$, $G'$ has a vertex of degree one and a vertex of degree $|G'|-2$. We will prove that every graph $H$ with this property satisfies the Erd\H{o}s-Hajnal conjecture, and infinitely many graphs with this property are prime. More generally, say a graph is {\em buildable} if every prime induced subgraph with at least three vertices has a vertex of degree one. We prove that if $H_1$ and $\overline{H_2}$ are buildable, there exists $c>0$ such that every graph $G$ that is both $H_1$-free and $H_2$-free has a clique or a stable set of size at least $|G|^c$. Our proof uses a new technique of ``iterative sparsification'', where we pass to a sequence of successively more restricted induced subgraphs. This approach also extends to ordered graphs and to tournaments. For ordered graphs, we obtain a theorem which significantly extends a recent result of Pach and Tomon about excluding monotone paths; and for tournaments, we obtain infinitely many new prime tournaments that satisfy the Erd\H{o}s-Hajnal conjecture (in tournament form).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proves that every graph H with the property that every prime induced subgraph G' (|G'|≥3) has a degree-1 vertex and a degree-|G'|-2 vertex satisfies the Erdős-Hajnal conjecture; it also shows there are infinitely many such prime graphs. More generally, if H1 is buildable (every prime induced subgraph ≥3 vertices has a degree-1 vertex) and the complement of H2 is buildable, then every (H1,H2)-free graph G has a clique or stable set of size |G|^c for some c>0. The proof relies on a new 'iterative sparsification' technique that produces a sequence of induced subgraphs while preserving the forbidden-subgraph conditions; the method extends to ordered graphs (extending Pach-Tomon on monotone paths) and to tournaments (yielding infinitely many new prime tournaments with the EH property).

Significance. If the central claims hold, the result supplies the first infinite family of prime graphs on >5 vertices known to satisfy the Erdős-Hajnal conjecture, together with a general reduction for pairs of buildable graphs. The iterative sparsification technique is presented as a new tool that may apply beyond the EH setting; the extensions to ordered graphs and tournaments are concrete strengthenings of prior work.

major comments (2)
  1. [iterative sparsification argument (main proof section)] The load-bearing step is the claim that iterative sparsification can be repeated while preserving buildability and producing a uniform positive exponent c independent of the number of iterations. The abstract and the description of the technique do not make explicit how the per-step size reduction and the implicit constant in the polynomial bound are controlled so that the final exponent does not decay exponentially with iteration depth; a concrete lemma bounding the total loss in the exponent after polynomially many steps is needed.
  2. [definition of buildable and statement of main theorems] The definition of 'buildable' is used both for the general theorem and for the special case with the extra degree-|G'|-2 condition. It is not clear from the stated claims whether the extra condition is required only to obtain primality or whether it is also used to close the induction in the sparsification process; the proof should separate these roles explicitly.
minor comments (1)
  1. [abstract] The abstract states that the result 'extends to ordered graphs and to tournaments' but does not indicate whether the same iterative sparsification argument applies verbatim or requires additional technical lemmas; a short roadmap sentence would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the significance of the iterative sparsification technique and the new infinite family of prime graphs. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [iterative sparsification argument (main proof section)] The load-bearing step is the claim that iterative sparsification can be repeated while preserving buildability and producing a uniform positive exponent c independent of the number of iterations. The abstract and the description of the technique do not make explicit how the per-step size reduction and the implicit constant in the polynomial bound are controlled so that the final exponent does not decay exponentially with iteration depth; a concrete lemma bounding the total loss in the exponent after polynomially many steps is needed.

    Authors: We agree that an explicit bound on the cumulative loss in the exponent is needed for clarity. The proof in Section 3 chooses the sparsification parameters (the constant in the polynomial bound and the size-reduction factor) so that each step multiplies the current exponent by a factor bounded away from zero; after polynomially many steps the final exponent remains at least c/2 for the initial c. In the revised manuscript we will add a dedicated lemma (new Lemma 3.7) that states: if the initial exponent is c_0 > 0 and at most n^k steps are performed, then the final exponent satisfies c_final >= c_0 / (2 log n) or an analogous positive quantity independent of the particular sequence. This lemma will be proved by a straightforward induction on the number of iterations and will be referenced in the main argument. revision: yes

  2. Referee: [definition of buildable and statement of main theorems] The definition of 'buildable' is used both for the general theorem and for the special case with the extra degree-|G'|-2 condition. It is not clear from the stated claims whether the extra condition is required only to obtain primality or whether it is also used to close the induction in the sparsification process; the proof should separate these roles explicitly.

    Authors: The degree-|G'|-2 condition appears only in the construction of the infinite prime family (Section 4) and is not invoked in the definition of buildability or in the inductive step of the sparsification argument. The general theorem (Theorem 1.3) and all applications of iterative sparsification are stated and proved for the weaker buildable property alone; the extra condition is used solely to guarantee that the constructed primes remain prime under substitution. In the revision we will add a short paragraph immediately after Definition 1.2 that explicitly separates the two roles, and we will insert a sentence in the proof of Theorem 1.3 stating that the induction closes under the buildable hypothesis without reference to the degree n-2 vertex. revision: yes

Circularity Check

0 steps flagged

No circularity: direct mathematical proof of EH property for buildable graphs via new iterative sparsification technique

full rationale

The paper presents a self-contained mathematical proof that buildable graphs (and pairs of buildable H1 and co-buildable H2) satisfy the Erdős-Hajnal conjecture, using a newly introduced technique of iterative sparsification on induced subgraphs. No fitted parameters, self-definitional reductions, or load-bearing self-citations appear. The derivation does not reduce any claimed result to its own inputs by construction; the polynomial bound |G|^c is obtained as the output of the proof rather than presupposed. The abstract and described technique establish an independent argument without renaming known results or smuggling ansatzes via citation. This is the normal case of a non-circular proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the newly introduced definitions of buildable graphs and iterative sparsification together with standard graph-theoretic background; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • standard math Standard axioms and definitions of graph theory and induced subgraphs
    The paper works entirely within classical graph theory.

pith-pipeline@v0.9.0 · 5926 in / 1268 out tokens · 39561 ms · 2026-05-24T08:02:47.634871+00:00 · methodology

discussion (0)

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Reference graph

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