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A directed Andr\'asfai-Erd\H{o}s-S\'os theorem and chromatic profiles of oriented cycles

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Pith's one-line read This paper proves a directed analogue of the classical minimum-degree coloring threshold: any digraph on $n$ vertices with no transitive tournament on $r$ vertices and minimum out-degree greater than $\frac{3r-7}{3r-4}n$ is…

desk verdict Nice directed AES package with a real gap in the central proof; the main theorem is not yet proven, but the paper is worth a careful referee. read the letter →

arxiv 2509.07760 v1 pith:KTIX75QA submitted 2025-09-09 math.CO

Yisai Xue This is my paper · ORCID
classification math.CO MSC 05C2005C3505C15
keywords chromaticprofiledigraphtransitivetournamentminimumout-degreeAndr\'asfai\textendashErd\H{o}s\textendashS\'ostheoremorientedcyclespentagonorientationscoloring
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines exact density thresholds for coloring digraphs that avoid a specified oriented subgraph. Its main result is a directed analogue of the classical Andr\'asfai\textendash Erd\H{o}s\textendash S\'os theorem: a digraph on $n$ vertices with no transitive tournament on $r$ vertices and minimum out-degree greater than $\frac{3r-7}{3r-4}n$ must be $(r-1)$-colorable, and no smaller density works. It also fixes the 2-coloring profile of every directed odd cycle at $1/2$, and of the three non-directed orientations of the pentagon at $1/3$. The significance is a precise dictionary between local density and global colorability in directed graphs, matching the known undirected picture.

What carries the argument

The key transfer device is the double orientation map, which sends each undirected edge to a pair of antiparallel arcs and turns the undirected sharpness example into a sharpness example for the directed theorem. The upper bound is carried by a saturation argument built around the 5-wheel-like digraph $\vec W_{r,t}$: two transitive tournaments on $r-2$ vertices sharing $t$ vertices, attached to vertices $v,w_1,w_2$ so that the only missing underlying edges are $vw_1$ and $vw_2$. Bounding the shared part $t\le r-3$ and counting how many common out-neighbors large sets must have forces $\delta^+(\hat D)\le \frac{3r-7}{3r-4}n$, contradicting the density assumption. For the pentagon orientations, the mechanism is a selection lemma that produces long chains of large common out-neighborhoods (Corollary 4.5), which is then used to embed arbitrary orientations of short cycles in dense digraphs.

What would settle it

A finite search for a $T_r$-free digraph with $\delta^+(D)>\frac{3r-7}{3r-4}n$ and chromatic number at least $r$ would refute the main theorem; none is expected. Because the pentagon upper bound depends on Lemma 4.4, an independent verification of that lemma for small $t$ and $\varepsilon$—or a counterexample where the common out-neighborhood fails to be linear—would settle whether the $1/3$ profile is truly established.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every $r\ge 3$, $\delta_\chi^+(T_r, r-1) = \frac{3r-7}{3r-4}$. In words, every $T_r$-free digraph with minimum out-degree above this fraction of $n$ is $(r-1)$-colorable, and the double orientation of the extremal graph used for the undirected theorem shows the constant cannot be lowered. The proof of the upper bound runs through a saturation argument: a maximal $T_r$-free digraph either has a complete multipartite underlying graph, in which case an induction on $r$ applies, or it contains a 5-wheel-like structure whose out-neighborhood counts contradict the density assumption. The same paper establishes the exact 2-coloring profiles for directed odd cycles ($1/2$) and for the three remaining pentagon orientations ($1/3$).

Load-bearing premise

For the pentagon results, the load-bearing assumption is an unproved selection lemma from an unpublished manuscript: among many large vertex sets, some fixed number always share a common out-neighborhood of linear size, with constants that do not degrade as $n$ grows; if that lemma fails, the upper bound showing profiles equal to $1/3$ loses its support.

Editorial extensions

If this is right

  • The undirected Andr\'asfai\textendash Erd\H{o}s\textendash S\'os theorem follows as a direct corollary by replacing each edge of a graph with a pair of antiparallel arcs.
  • A stability statement accompanies the main theorem: a $T_r[t]$-free digraph with minimum out-degree at least $\frac{3r-7}{3r-4}+\varepsilon$ times $n$ can be made $(r-1)$-partite by deleting $o(n^2)$ arcs.
  • The directed odd-cycle result means any digraph with minimum out-degree above $n/2$ and chromatic number at least $3$ must contain a directed odd cycle of every specified length.
  • For the pentagon, the identical $1/3$ profile for the three non-directed orientations says that above one-third density, avoiding any one of those orientations forces bipartiteness.
  • If the main theorem is correct, it gives a complete directed analogue of the classical minimum-degree-to-chromatic-number hierarchy for transitive tournaments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's statements, the $1/3$ threshold for the pentagon orientations suggests that other orientations of odd cycles admitting a homomorphism to the directed triangle may also have profile $1/3$; the paper leaves this open in Question 2.
  • The lower-bound constructions all contain source vertices, so a natural test not performed here is whether the thresholds drop when minimum out-degree is replaced by minimum semi-degree.
  • If the unproved selection lemma were given explicit constants, the stability theorem's $o(n^2)$ arc-deletion bound might become quantitative, yielding a concrete exponent in the arc-deletion count.
  • The contrast the paper records between chromatic profiles and chromatic thresholds for the pentagon orientations suggests that profile equality need not track threshold ordering, a phenomenon worth testing on longer odd cycles.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The manuscript studies the chromatic profile δ^+_χ(H,k) for digraphs. The main result, Theorem 1.1, determines the profile for transitive tournaments T_r: δ^+_χ(T_r, r−1) = (3r−7)/(3r−4), with sharpness given by double orienting an AES-type graph (the C_5 blow-up joined with K_{r−3} for r≥4, and the C_5 blow-up alone for r=3). Theorem 1.3 gives δ^+_χ(\vec{C}_{2ℓ+1},2)=1/2 for every ℓ≥1. Theorem 1.4 gives δ^+_χ(C_5',2)=δ^+_χ(C_5'',2)=δ^+_χ(C_5''',2)=1/3. Section 2 proves the upper bound via a saturation argument using 5-wheel-like digraphs; Section 4 uses a selection lemma, embedding lemmas, and an odd-girth case split; Section 5 applies Theorem 1.1 through the diregularity lemma to obtain a stability result.

Significance. If completed, the paper is a substantial contribution to extremal digraph theory. Theorem 1.1 is an exact directed analogue of the Andrásfai–Erdős–Sós theorem with an explicit sharp construction, and Theorem 1.4 resolves the two-coloring profiles for all three non-directed pentagon orientations. The stability application in Theorem 1.2 is a natural and plausible consequence. The constructions and the general strategy are clear and well motivated. However, the proof of Theorem 1.1 contains an unproved structural step in Claim 2.10, and the proof of Theorem 1.4 relies on a selection lemma cited to an in-preparation manuscript; both issues are load-bearing and need to be addressed before the central claims can be considered established.

major comments (3)
  1. [Section 2, Claim 2.10] The step "Let Q_i' = Q_i − y_i + x. Then Q_1' and Q_2' are transitive tournaments on r−2 vertices" is not justified. The definition of X only guarantees that every vertex of Q_1∩Q_2 dominates x; for a private vertex q in V(Q_i)\V(Q_{3−i}) with q≠y_i, the manuscript supplies no arc between q and x and no orientation consistent with the transitive order of Q_i. The condition that y_i does not dominate x only excludes the arc y_i→x; it does not place x at the position of y_i in the transitive order. If x→q for some private q that precedes y_i, or if x is non-adjacent to such a q, then Q_i' need not be a transitive tournament. Since the maximality of t is then applied to Q_1' and Q_2' to conclude d^+(W,x)≤|W|−3, and this bound is exactly what makes inequality (5) contradict the minimum-out-degree assumption, the upper-bound proof of Theorem 1.1 is incomplete at this point. A full proof of the transitivity of Q_i' (or a different argument for d^+(W,x)≤|W|−3) is required.
  2. [Section 4, Lemmas 4.4–4.6] Lemma 4.4 is stated as a result of Gao–Liu–Wu–Xue and cited to the in-preparation manuscript [11], with no proof given. This lemma is load-bearing: Corollary 4.5 applies it iteratively to construct the disjoint common out-neighbourhood sets X_1,...,X_k, and Lemma 4.6 uses those sets to embed arbitrary oriented cycles in digraphs with minimum out-degree at least (1/2+ε)n. Lemma 4.6 is in turn used in Lemma 4.7(iii) and in Case 1 of Theorem 4.11. As submitted, the upper-bound proof of Theorem 1.4 depends on an unverifiable premise from an unpublished source. The authors should either include a complete proof of Lemma 4.4 or replace it with a publicly available reference that contains the proof.
  3. [Section 4, Theorem 4.11, Case 2] In the odd-girth-five case, after Claim 4.12 the proof constructs C_5^1 explicitly, but for C_5^2 and C_5^3 it only says the argument is analogous and refers to Figure 4.6. Since Theorem 1.4 requires the upper bound δ≤1/3 for all three orientations, the omission is not merely cosmetic: the replacement sequences for the two remaining orientations must be specified and proved to produce oriented 5-cycles at every intermediate step and the required forbidden pentagon at the end. Please provide the full details for these two cases.
minor comments (3)
  1. [Section 2, proof of Claim 2.10] The sentence "for each x∈X, there exists y_i ∈ ({w_i}∪V(Q_i))\V(Q_{3−i}) that does not dominate x" is ambiguous because it suggests a single vertex, while the subsequent argument needs two distinct vertices y_1 and y_2 with y_1∈V(Q_1)\V(Q_2) and y_2∈V(Q_2)\V(Q_1). Please state the choice of both vertices explicitly.
  2. [Section 4, Lemma 4.6] The phrase "P=C−v is an oriented path of length k−1" is inconsistent with the usual meaning of length as the number of arcs; since C has k vertices, P has k−1 vertices and k−2 arcs. Please rephrase as "on k−1 vertices".
  3. [Section 5, proof of Theorem 1.2] The notation "Fix 1≫γ≫d≫ε>0" should be spelled out as a hierarchy of constants, and the object called the pε,dq-reduced graph R should be called a reduced digraph for consistency.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 1.1 and Theorem 1.3 are self-contained; the pentagon-profile upper bound relies on an unproved lemma cited to the author's own in-preparation manuscript.

  1. self citation load bearing [Section 4, Lemma 4.4; used via Corollary 4.5 and Lemma 4.6 in Lemma 4.7(iii) and Theorem 4.11 (proof of Theorem 1.4 upper bound)]
    "Lemma 4.4 (Gao, Liu, Wu and Xue, [11]). Let tPN and εPp0,1q. Then there exist α=αpε,tq and m=mpε,tq such that for any V1,V2,¨¨¨,VmĎrnseach with size at least εn, there exist 1ďi1ăi2㨨¨ăitďmwith|Şt j“1Vij|ěαn."

    Theorem 1.4's upper bound (Theorem 4.11) uses Lemma 4.6, and Lemma 4.6's proof invokes Corollary 4.5, obtained by 'Iterating the above proposition' from Lemma 4.4. Lemma 4.4 is not proved in this paper; it is quoted from [11], an in-preparation manuscript whose author list includes the present author. The pentagon-profile result is therefore supported at a load-bearing point by an unverified self-citation rather than by a self-contained derivation. The main Theorem 1.1 and Theorem 1.3 are independent of this lemma, so the circularity is partial rather than total.

full rationale

I found no self-definitional or fitted-input circularity. Theorem 1.1's lower bound is an explicit double-oriented construction and its upper bound is a saturation argument using Lemmas 2.2/2.5 and Claims 2.9–2.10; no constant is fitted from the conclusion. Theorem 1.3 uses an explicit family and a direct directed-cycle lemma. The lower bounds in Theorem 1.4 are explicit constructions, and the upper-bound argument is an extremal proof except for its reliance on Lemma 4.4. The skeptical note about Claim 2.10 is a possible correctness gap (an unproved claim about replacing a vertex in Q_i), not a circularity: it does not define the conclusion into the assumptions, so it does not affect this score. The only circularity-adjacent step is the load-bearing citation of the author's own in-preparation Lemma 4.4.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No invented entities or fitted parameters. The paper's constants are exact and arise from explicit constructions. The main external debt is the unpublished selection lemma [11], which supports the pentagon upper bound, plus standard regularity and embedding tools for the stability application.

assumptions (4)
  • ad hoc to paper Lemma 4.4 (Gao-Liu-Wu-Xue selection lemma): among m sufficiently large subsets of [n] each of size at least εn, there are t with common intersection at least αn.
    Stated without proof and cited to unpublished manuscript [11]; used to prove Corollary 4.5, Lemma 4.6, and hence the upper bound of Theorem 1.4.
  • domain assumption Degree form of the Diregularity Lemma (Lemma 5.1) from Taylor [25] and Keevash-Kühn-Osthus [18].
    Standard regularity lemma for digraphs; used in the proof of Theorem 1.2, stated with reference.
  • domain assumption Embedding Lemma for digraphs, invoked as a standard result in the proof of Theorem 1.2.
    Not stated or referenced precisely; needed to transfer T_r[r t]-freeness from D to the reduced digraph R.
  • standard math Known fact that every oriented cycle that is not a directed cycle contains a sink vertex.
    Used in Lemma 4.6 to reduce to a path embedding; elementary and correct.

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Pith. "Pith review of A directed Andr\'asfai-Erd\H{o}s-S\'os theorem and chromatic profiles of oriented cycles." pith.science (2026). https://pith.science/paper/KTIX75QA

@misc{pith2026250907760,
  author       = {Pith},
  title        = {Pith review of: A directed Andr\'asfai-Erd\Hos-S\'os theorem and chromatic profiles of oriented cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTIX75QA}},
  note         = {Machine review of arXiv:2509.07760}
}
abstract

The chromatic profile of a digraph $H$, denoted by $\delta_{\chi}^{+}(H,k)$, is the infimum $d$ such that any $H$-free digraph $D$ on $n$ vertices with minimum out-degree $\delta^{+}(D) \ge dn$ must be $k$-colorable. We determine the exact chromatic profile for several fundamental classes of digraphs. Our main result is a directed analogue of the Andr\'asfai-Erd\H{o}s-S\'os theorem, stating that $\delta_\chi^{+}(T_r, r-1)=\frac{3 r-7}{3 r-4}$, where $T_r$ is the transitive tournament on $r$ vertices. We then determine the chromatic profile for directed odd cycles, showing that $\delta^+_\chi(\overrightarrow{C}_{2\ell+1},2)=1/2$ for all $\ell\ge 1$. Finally, we resolve the profile for the three remaining orientations of the pentagon, establishing that $\delta_{\chi}^{+}(C_{5}',2)=\delta_{\chi}^{+}(C_{5}'',2)=\delta_{\chi}^{+}(C_{5}''',2)=1/3$.

Figures

Figures reproduced from arXiv: 2509.07760 by the authors.

Figure 1.1
Figure 1.1. Orientations of C5 Finally, we investigate the chromatic profile for orientations of the pentagon C5. There are four non-isomorphic orientations of the pentagon, depicted in [PITH_FULL_IMAGE:figures/full_fig_p002_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. The 5-wheel-like digraph ÝÑWr,t. We are now ready to prove Theorem 2.7. We use a saturation argument, dividing the proof into two cases based on whether the underlying graph of the saturated digraph is complete multipartite. Theorem 2.7. Let D be a Tr-free digraph on n vertices. If δ `pDq ą 3r ´ 7 3r ´ 4 n, then D is pr´1q-colorable. Proof. Consider the saturated digraph Dˆ of D, which is obtained by adding arcs unt… view at source ↗
Figure 2.2
Figure 2.2. Lower bound for Andr´asfai-Erd˝os-S´os theorem. [PITH_FULL_IMAGE:figures/full_fig_p006_2_2.png] view at source ↗
Figures from the paper (6 more)
Figure 4.1
Figure 4.1. Figure 4.1: Illustrations of the constructions An, Bn, and ÝÑC3rn1, n2, n3s. Construction 3 (For C 1 5 , see [PITH_FULL_IMAGE:figures/full_fig_p007_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Find C 2 5 starting from an arbitrary triangle. (ii) Existence of C 1 5 . From part (i), we know D contains a C 2 5 . Let its vertices be tv1, . . . , v5u with arcs forming paths v1 Ñ v2 Ñ v3 Ñ v4 and v1 Ñ v5 Ñ v4. If |N `pv1q X N `pv4q| ě 3, a C 1 5 is easily found.…
Figure 4.3
Figure 4.3. Figure 4.3: Find C 1 5 starting from a C 2 5 . (iii) Existence of C 3 5 . We proceed by contradiction, assuming D is C 3 5 -free. First, there must exist an arc pu1, u2q with |N `pu1q X N `pu2q| ď 2. Otherwise, the two endpoints of every arc have at least three common out-neighb…
Figure 4.4
Figure 4.4. Figure 4.4: Find C 3 5 . Define: U1 “ N `pu1qzN `ru2s, U2 “ N `pu2qzN `ru1s, U3 “ V pDqzpU1 Y U2 Y tu1, u2uq, where N `ruis “ N `puiq Y tuiu. Then |U1|, |U2| ě p1{3 ` εqn ´ 3 and |U3| ă p1{3 ´ 2εqn. Claim 4.8. For any w P V pDqztu1, u2u, exactly one of N `pwqXU1 and N `pwqXU2 is…
Figure 4.5
Figure 4.5. Figure 4.5: Find C 1 5 starting from an arbitrary 5-cycle. By iteratively applying Claim 4.12, we can also find C 2 5 and C 3 5 in D. The argument is analogous to the one used to obtain C 1 5 , so we omit the details and illustrate the construction in [PITH_FULL_IMAGE:figures/f…
Figure 4.6
Figure 4.6. Figure 4.6: Find C 2 5 and C 3 5 starting from an arbitrary 5-cycle. Case 3. goddpDq ě 7. Let C be a minimum odd cycle in GpDq. Then C is an induced cycle and |C| ě 7. Moreover, d `pV pCq, vq ď 2 for every v P V pDq; otherwise, there exists an odd cycle of length smaller than |C…

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