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Phase transition of degenerate Tur\'{a}n problems in $p$-norms

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Pith's one-line read The p-norm Turán number of a degenerate hypergraph family switches sharply at p = 1/(r-1-α), with a tight star constant above and pseudorandom growth below.

desk verdict Theorem 1.2 is false for r≥3: the large-p bound should be n^{p(r-1)}, not (n/(r-1))^p, and the proof repeats the same dimensional error in Claim 4.4. read the letter →

arxiv 2411.15579 v3 pith:YYWYSEL3 submitted 2024-11-23 math.CO

Jun Gao , Xizhi Liu , Jie Ma , Oleg Pikhurko This is my paper · ORCID
classification math.CO MSC 05C3505C6505D40
keywords degenerateTuránproblemdegreepowersp-normphasetransitionhypergraphspartitionnumberevencyclesregularization
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

This paper proves a sharp phase transition for p-norm Turán numbers of degenerate families of r-uniform hypergraphs. For a forbidden family F whose ordinary Turán number grows at most like $n^{1+\alpha}$ with $\alpha < r-1$, the maximum p-norm of an F-free r-graph is $O(n^{1+p\alpha})$ when $p$ is below the threshold $1/(r-1-\alpha)$, and at most $(\tau_{\mathrm{part}}(F)-1+o(1))(n/(r-1))^p$ above it. In the graph case $r=2$, the supercritical bound is asymptotically tight because the partition number $\tau_{\mathrm{part}}(F)$ equals the independent covering number $\tau_{\mathrm{ind}}(F)$. The paper also removes the logarithmic factor conjectured removable by Füredi and Kündgen at the threshold for several families, including short even cycles and s-bounded bipartite graphs. These results matter because the p-norm interpolates between edge count and maximum degree, so the threshold describes how extremal constructions change as the objective shifts from counting edges to concentrating them.

What carries the argument

The load-bearing tool is a p-norm version of the classical $\Delta$-almost-Regularization Theorem of Erdős–Simonovits, stated as Lemma 3.1. Given an r-graph with p-norm at least $C n^{1+p\alpha}$, this lemma produces a subgraph $H$ on $m$ vertices that keeps a $(1-\varepsilon)$ fraction of the p-norm, has maximum degree at most a constant times $(\|H\|_p/m)^{1/p}$, and has more than $\hat{C} m^{1+\alpha}$ edges. This converts an F-free graph with overly large p-norm into an F-free graph with too many edges, contradicting the assumed bound $\mathrm{ex}(n,F)=O(n^{1+\alpha})$. In the supercritical regime, the proof instead uses a semibipartite upper bound for complete r-partite hypergraphs (Proposition 2.9 from [HHL+23]) together with the partition number $\tau_{\mathrm{part}}(F)$, and it bootstraps the subcritical result at a smaller exponent $\hat{p}<p$. The star-like hypergraph $S_r(n,t) = \{ e \in \binom{[n]}{r} : |e \cap [t]| = 1 \}$ supplies the matching lower bound with $t = \tau_{\mathrm{part}}(F)-1$.

What would settle it

Compute the p-norm Turán number for the 3-graph F from Section 7 with $\tau_{\mathrm{ind}}(F)=3$ and $\tau_{\mathrm{part}}(F)=4$, for $p>2$. If F-free 3-graphs on $n$ vertices can have p-norm exceeding $(3+\varepsilon)(n/2)^p$ infinitely often, the supercritical bound of Theorem 1.2 is false; the open question is whether the star lower bound $(\tau_{\mathrm{ind}}(F)-1+o(1))(n/2)^p$ can be pushed closer to the $\tau_{\mathrm{part}}(F)-1$ upper bound.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1.2: if F is a degenerate family of r-graphs with $\mathrm{ex}(n,F)=O(n^{1+\alpha})$ for some constant $\alpha \in [r-2, r-1)$, then for every $p>1$ there is a constant $C_F$ with $\mathrm{ex}_p(n,F) \le C_F n^{1+p\alpha}$ for $1<p<1/(r-1-\alpha)$, and $\mathrm{ex}_p(n,F) \le (\tau_{\mathrm{part}}(F)-1+o(1))(n/(r-1))^p$ for $p>1/(r-1-\alpha)$. Here $\tau_{\mathrm{part}}(F)$ is the minimum, over r-partite members of F, of the smallest class size in a partition of the vertex set into r classes each met by every edge exactly once. The two regimes correspond to pseudorandom, almost regular constructions below the threshold and to star-like hypergraphs $S_r(n, \tau_{\mathrm{part}}(F)-1)$ above it. For $r=2$, the parameters $\tau_{\mathrm{ind}}$ and $\tau_{\mathrm{part}}$ coincide, so the paper obtains the exact leading coefficient in the star regime, going beyond the earlier proof of Füredi and Kündgen. The paper also establishes Theorem 1.3, a general $O(n^{p^*(r-1)}\log n)$ bound at the threshold $p^*=1/(r-1-\alpha)$, and Theorem 1.4, which removes the logarithmic factor for families of short even cycles and for s-bounded bipartite graphs.

Load-bearing premise

The theorem assumes that the ordinary Turán number of the forbidden family is polynomially bounded as $O(n^{1+\alpha})$ with $\alpha < r-1$; for many natural families, such as long even cycles, this polynomial exponent is still open, so the phase-transition conclusion applies only to families for which that external bound has been proved.

Editorial extensions

If this is right

  • For any degenerate family F satisfying $\mathrm{ex}(n,F)=O(n^{1+\alpha})$, the p-norm Turán number is determined up to a constant factor in the subcritical regime and asymptotically in the supercritical regime, for every $p>1$.
  • In the graph case, the exact leading constant $\mathrm{ex}_p(n,F)=(\tau_{\mathrm{ind}}(F)-1+o(1))n^p$ for $p>1/(1-\alpha)$ follows from the theorem together with the star construction.
  • The threshold bound at $p=p^*$ extends Füredi–Kündgen's log-factor conjecture to all uniformity ranks $r$, and the families in Theorem 1.4 now satisfy the conjectured bound without the logarithmic factor.
  • For $\{C_4,\dots,C_{2\ell}\}$, the bound $\mathrm{ex}_{\ell/(\ell-1)}(n,\{C_4,\dots,C_{2\ell}\}) \le 765 n^{\ell/(\ell-1)}$ holds, and for s-bounded bipartite F, $\mathrm{ex}_s(n,F) \le 2(|V(F)|^s/s! + |V(F)|) n^s$ holds.
  • If $\mathrm{ex}(n,F)=O(n^{1+\beta})$ with $\beta \le r-2$, the theorem implies the star-like bound $(\tau_{\mathrm{part}}(F)-1+o(1))(n/(r-1))^p$ for every $p \ge 1$, so no subcritical regime exists.

Reading between the lines

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

  • If the Erdős–Simonovits Rational Exponent Conjecture holds for a graph family, the subcritical exponent $n^{1+p\alpha}$ is tight, so the phase transition would give a complete piecewise-linear description of the p-norm exponent in terms of the ordinary Turán exponent.
  • For $r \ge 3$, the gap between $\tau_{\mathrm{ind}}(F)$ and $\tau_{\mathrm{part}}(F)$ leaves open whether the supercritical limit exists; the paper's own Problem 7.1 asks exactly this, and the 3-graph example with $\tau_{\mathrm{ind}}=3$, $\tau_{\mathrm{part}}=4$ is the natural first test case.
  • The p-norm regularization lemma is a standalone tool that could be applied to other degree-based objectives, such as counting copies of a fixed subgraph or studying the $(t,p)$-norm Turán numbers mentioned in the concluding remarks, by replacing the edge-count condition with the relevant count.
  • A concrete testable prediction of the phase transition: for complete bipartite graphs $K_{s,t}$ with $t$ large, where $\mathrm{ex}(n,K_{s,t})=\Theta(n^{2-1/s})$, the p-norm extremal number should be $\Theta(n^{1+p(1-1/s)})$ for every $1<p<s$.
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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 / 2 minor

Summary. The paper studies the p-norm Turán number ex_p(n,F), the maximum of the sum of p-th powers of vertex degrees over F-free r-graphs on n vertices. It claims a phase transition for degenerate families F satisfying ex(n,F)=O(n^{1+alpha}): for 1<p<1/(r-1-alpha) it gives ex_p(n,F)=O(n^{1+p alpha}), while for p>1/(r-1-alpha) it claims ex_p(n,F) is at most (tau_part(F)-1+o(1))(n/(r-1))^p. At the threshold it proves an O(n^{p(r-1)} log n) bound, and it confirms the conjectured removal of the log factor for several bipartite graph families, including short even cycles and one-side degree-bounded graphs. The proofs use a p-norm regularization lemma, semibipartite reductions, and dependent random choice. The graph case r=2 is also re-proved with an improved constant.

Significance. If the main theorem were correct, it would be a substantial extension of the Füredi-Kündgen graph result to hypergraphs and would settle the log-factor conjecture at the threshold for several natural graph families. The paper contains potentially useful tools, especially the p-norm regularization lemma and the explicit bounds in Theorem 1.4. However, the central hypergraph statement is false: Theorem 1.2 as stated is contradicted by an elementary construction for r>=3. The advertised hypergraph phase transition therefore cannot stand, and the remaining graph-case results, though possibly valid, are a much smaller contribution than the paper claims.

major comments (3)
  1. [Theorem 1.2 / Proposition 4.2] Theorem 1.2 is false for r>=3. Let F={abc, def} be the 3-graph consisting of two disjoint triples. F is 3-partite, hence degenerate, and ex(n,F)=Theta(n^2) for n large, since the largest intersecting 3-graph has about C(n-1,2) edges. Thus the hypothesis holds with r=3 and alpha=1, and tau_part(F)=2. For p=2>1/(r-1-alpha)=1, the theorem predicts ex_2(n,F) is at most (1+o(1))(n/2)^2 ~ n^2/4. But the star S_3(n,1), consisting of all triples containing vertex 1, is F-free and has p-norm C(n-1,2)^2+(n-1)(n-2)^2 ~ n^4/4, so ex_2(n,F)=Omega(n^4). Even the paper's own Corollary 2.5, Eq. (1), gives ex_2(n,F) >= n(3 ex(n,F)/n)^2 = Omega(n^3), already contradicting the claimed O(n^2) upper bound. The introduction's heuristic predicting n^{p(r-1)} in the large-p regime points to the same conclusion, so the error is not merely a typo in one constant.
  2. [Claim 4.4] The proof of the large-p regime invokes Proposition 2.9 in the wrong direction. The text says that because F is contained in the complete r-partite r-graph K^r_{s1,...,sr} and S is F-free, Proposition 2.9 bounds |S|. This implication is backwards: if F is a subgraph of K, then every K-free graph is F-free, so an upper bound for K-free graphs does not upper-bound F-free graphs. For the family F={abc, def} from the previous comment, F is a proper subgraph of K^3_{2,2,2}, and F-free semibipartite graphs can have far more edges than K^3_{2,2,2}-free graphs; for example, a star with one high-degree vertex in U is F-free. Consequently the O(n) bound on the sum of degrees of high-degree vertices in Eq. (10) is unjustified.
  3. [Claim 4.4] Even granting Proposition 2.9, the absorption of error terms in Claim 4.4 is not justified in the stated parameter range. The assertion |S2| <= |U|^2 C(n,r-2) <= (epsilon/(6r))(n/(r-1))^p requires 2 delta_2 + r - 2 < p. The paper only guarantees delta_2 <= (p-1)/p, which does not imply this inequality; for instance, with r=4, alpha=2, and p=1.5, one has p>1/(r-1-alpha)=1, but any positive delta_2 gives 2 delta_2 + 2 > 1.5. The same obstruction occurs in the absorption of the first term of Proposition 2.9 into (epsilon/2)n/(r-1), since its exponent r-1-1/(s1...s_{r-1})+delta_2 is not always less than 1. Thus the proof of Proposition 4.2 does not establish the claimed upper bound even if the direction of the Proposition 2.9 application were fixed.
minor comments (2)
  1. [Claim 6.8] In Claim 6.8 the inequality W_{\ell+1}(H) >= 4^{\ell-1} n^2 is obtained by replacing v(H) by n in the denominator, but the proof only gives v(H) <= 2n. This replacement is not valid and overestimates the constant by a factor 2^{\ell-2}. The argument still works if one replaces n^{\ell-2} by (2n)^{\ell-2}, yielding the weaker but sufficient bound W_{\ell+1}(H) >= 2^{\ell} n^2, so this is a harmless constant slip.
  2. [Fact 1.1] The second lower bound in Fact 1.1 is not the true p-norm of the star-like r-graph S_r(n,t) when r>=3. For r=3, each center vertex has degree about n^2/2, so the p-norm is about t (n^2/2)^p, whereas the displayed expression (n/(r-1))^p is only t (n/2)^p. The displayed inequality is true but far weaker than the actual star construction, and the surrounding text should not attribute it to the p-norm of S_r(n,t).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorem is derived from the stated growth-rate assumption via an internal p-norm regularization lemma and external Turán-type estimates; the single co-authored citation is a general lemma, not an encoding of the target result.

full rationale

Walking the derivation chain of Theorems 1.2, 1.3, and 1.4 discloses no circular step. The central upper bound for the regime p < 1/(r-1-alpha) is obtained in Proposition 4.1 by applying the internally proved Lemma 3.1 (the p-norm regularization) to an assumed counterexample and contradicting the assumed ordinary Turán bound ex(n,F)=O(n^{1+alpha}); no parameter is fitted and no conclusion is used as its own hypothesis. The regime p > 1/(r-1-alpha) in Proposition 4.2 invokes Proposition 4.1 for a lower value of p to obtain a contradiction; this is a legitimate bootstrap from an already-proven statement, not circularity. The only citation to prior work with overlapping authorship is Proposition 2.9, quoted from [HHL+23] (which includes coauthor X. Liu); it is a general, parameter-free upper bound on ex(m,n,K^r_{s1,...,sr}) whose assumptions do not include the target result, so under the stated criteria it counts as independent support rather than circular self-citation. The threshold 1/(r-1-alpha) is a function of the assumed exponent alpha, not an output used to define alpha, so there is no fitted-input-called-prediction pattern. The graph-level results in Theorem 1.4 rest on external theorems (Füredi, Lam-Verstraëte, Naor-Verstraëte, Erdős-Simonovits, Sağlam) and internal walk-count estimates. The paper's own remark that parts of the r=2 case overlap with [FK06] is an attribution, not a circular derivation, and Theorem 7.5 is only sketched but is presented as a side extension, not as support for the main theorems. A possible algebraic slip in Claim 4.4 for r>=3, where binom(n,r-1) is silently replaced by n/(r-1), would be a correctness concern if it stood, but it is not a circularity: it does not make the claimed conclusion identical to an input. Overall the paper is self-contained against external benchmarks in the sense relevant to the circularity pass.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's results rest on standard extremal combinatorics tools and on external growth-rate bounds for specific families. No free parameters are fitted to data; the constants in the theorems are existential or explicit but not calibrated. The central assumption is the polynomial upper bound ex(n,F)=O(n^{1+α}), plus standard theorems used as black boxes (Erdős's degenerate bound, Füredi's s-bounded bound, Lam-Verstraëte, Naor-Verstraëte, Erdős-Simonovits, Sağlam, dependent random choice). One internal result from the authors' own prior work (Proposition 2.9 from [HHL+23]) is used as a tool; it is not the target result and is cited normally.

assumptions (5)
  • domain assumption F is a degenerate family of r-graphs (contains an r-partite r-graph).
    Throughout the paper; this guarantees ex(n,F)=o(n^r) and enables the growth-rate assumption.
  • domain assumption ex(n,F)=O(n^{1+α}) for some α in [r-2,r-1) (or α ≥ r-2 in Theorem 1.3).
    The central precondition of Theorems 1.2 and 1.3; the threshold and bounds depend on α.
  • standard math Theorem 2.7 (Erdős): a degenerate family satisfies ex(n,F)=O(n^{r-δ}).
    Used in Proposition 4.2 to bound ex(n,F1) for the (r-1)-graph F1.
  • standard math Proposition 2.9 from [HHL+23] bounding ex(m,n,K^r_{s1,...,sr}).
    Used in Proposition 4.2 to bound the semibipartite subgraph S.
  • standard math Theorems 6.1, 6.2, 6.3, 6.9 (Lam-Verstraëte, Naor-Verstraëte, Erdős-Simonovits/Sağlam, Füredi-Naor-Verstraëte).
    External bounds and walk inequalities used in the proof of Theorem 1.4.

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Pith. "Pith review of Phase transition of degenerate Tur\'{a}n problems in $p$-norms." pith.science (2026). https://pith.science/paper/YYWYSEL3

@misc{pith2026241115579,
  author       = {Pith},
  title        = {Pith review of: Phase transition of degenerate Tur\'an problems in $p$-norms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYWYSEL3}},
  note         = {Machine review of arXiv:2411.15579}
}
abstract

For a positive real number $p$, the $p$-norm $\left\lVert G \right\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices. F\"{u}redi and K\"{u}ndgen \cite{FK06} show that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions, assuming a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. The main contribution of our paper is extending this result to hypergraph. Moreover, in the case of graph, our proof differs from that in \cite{FK06}, offering the advantage of producing the correct constant factor when $p > p_{F}$. When $p = p_F$, F\"{u}redi and K\"{u}ndgen proved a general upper bound on $\mathrm{ex}_{p}(n,F)$, tight up to a $\log n$ factor, and conjectured that this factor is unnecessary. We confirm this conjecture for several well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.

Figures

Figures reproduced from arXiv: 2411.15579 by the authors.

Figure 1
Figure 1. Exponents of exp(n, K3,3), exp(n, C4), and exp(n, C6). For a family F of r-graphs, we define τpart(F) := min {τpart(F): F ∈ F is r-partite} and τind(F) := min {τind(F): F ∈ F is r-partite} . Theorem 1.2. Let r ≥ 2 be an integer and p > 1 be a real number. Suppose that F is a degenerate family of r-graphs satisfying ex(n, F) = O(n 1+α) for some constant α ∈ [r − 2, r − 1). Then there exists a constant CF > 0 such tha… view at source ↗

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Forward citations

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