Concentration of the maximum size of an induced subtree in moderately sparse random graphs

Juan Carlos Buitrago Oropeza

REVIEW 2 major objections 4 minor 26 references

Concentration of the maximum size of an induced subtree in moderately sparse random graphs

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For edge probabilities down to $n^{-(e-2)/(3e-2)+\varepsilon}$, the largest induced tree in $G(n,p)$ is almost surely one of two consecutive integers.

desk verdict A genuine extension of two-point concentration for induced trees into the sparse regime, with the main risk being imported bounds from [21] that the paper does not reproduce. read the letter →

arxiv 2506.02801 v2 pith:P6SQNPPO submitted 2025-06-03 math.CO

Juan Carlos Buitrago Oropeza This is my paper

classification math.CO MSC 05C8005C0560C05

keywords binomialrandomgraphinducedtreetwo-pointconcentrationsecondmomentmethodsparsegraphsmaximumsubgraph

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 proves that two-point concentration for the largest induced tree, previously known when the edge probability $p$ is constant, persists in the moderately sparse regime $n^{-(e-2)/(3e-2)+\varepsilon} \le p = o(1)$. Concretely, there is a constant $\delta$ such that asymptotically almost surely the maximum size of an induced subtree in $G(n,p)$ is either $\lfloor 2\log_{1/(1-p)}(enp)+\delta\rfloor$ or that value plus one. The rough size $2\log_{1/(1-p)}(enp)$ was already known; the new content is that the random graph cannot miss this size class by more than one step. The proof shows that, at the critical scale $k$, the second moment of the number of induced $k$-vertex trees is within a factor $1+o(1)$ of its square, so Chebyshev's inequality forces such a tree to appear with probability tending to one.

What carries the argument

The argument runs on the second moment of $X_k$, the number of induced $k$-vertex trees. The governing inequality is $\operatorname{Var} X_k/(\mathbb{E}X_k)^2 \le \sum_{\ell=2}^{k-1} F_\ell/(\mathbb{E}X_k)^2$, where $F_\ell$ counts pairs of $k$-trees sharing exactly $\ell$ vertices, weighted by their edge overlap. The paper partitions the range of $\ell$ into four intervals and forces each piece of the sum to be $o(1)$. Small overlaps use the crude bound $N(k,\ell,r)\le k^{2(k-2)}$; medium overlaps use $f(k,\ell,r)$, the maximum over forests $F$ with $r$ edges on $\ell$ vertices of the number of $k$-trees that induce $F$, with bounds imported from the constant-$p$ proof; the near-complete-overlap range is handled by sharpening this via the rooted-forest enumeration formula $\varphi(\ell,r)$ and by ratio tests such as $\hat I_{\ell+1}/\hat I_\ell$. The exponent $(e-2)/(3e-2)$ emerges from the ratio test in the last interval $\ell \in (k-1/(2p), k-1)$.

What would settle it

At the boundary $p = n^{-(e-2)/(3e-2)}$, compute the sum $\sum_{\ell=2}^{k-1} F_\ell/(\mathbb{E}X_k)^2$ with $k=\lfloor 2\log_{1/(1-p)}(enp)+\delta\rfloor$; the paper's own ratio test in Section 3.4 marks this as the threshold where the final interval's contribution stops being $o(1)$, so a direct evaluation showing the ratio stays bounded away from zero would refute the stated range.

Watch

Extended reading notes

Core claim

At the center is Theorem 1.7: for every $\varepsilon>0$, if $n^{-(e-2)/(3e-2)+\varepsilon} \le p = o(1)$, then the maximum size of an induced tree in the binomial random graph $G(n,p)$ is concentrated on two consecutive values, $\{g(n), g(n)+1\}$, where $g(n)=\lfloor 2\log_{1/(1-p)}(enp)+\delta\rfloor$ for some constant $\delta>0$. An induced tree is a vertex set whose induced subgraph is a tree. This is the sparse extension of the constant-$p$ theorem: even when $p$ decays as a small power of $1/n$, the largest tree-shaped induced subgraph is almost surely pinned to within one of a deterministic function. The proof locates the main difficulty in the overlap of large candidate trees: pairs of $k$-vertex trees sharing almost all of their vertices, $\ell \ge k - O((\ln k)/p)$, need new estimates because the previously available bounds on extensions of a fixed forest are too weak there.

Load-bearing premise

The load-bearing premise is a previously established bound on how many large trees can contain a fixed smaller forest, asserted to hold for every edge probability $p\in(0,1)$; if that bound degrades in the sparse range where the forest contains almost all vertices of the tree, the variance estimate (11) fails.

Editorial extensions

If this is right

In the stated range, the largest induced tree is pinned to within one of a deterministic value: $g(n)$ or $g(n)+1$, with $g(n)=\lfloor 2\log_{1/(1-p)}(enp)+\delta\rfloor$.
The previously known first-order asymptotics for the maximum induced tree is upgraded to genuine concentration for every $p$ between $n^{-(e-2)/(3e-2)+\varepsilon}$ and any $p=o(1)$.
Because the maximum is monotone in $k$, the Markov upper bound also excludes induced trees of size $g(n)+2$ or larger, not only those above $g(n)+1$.
The variance control at the critical scale is quantitative: $\operatorname{Var} X_k/(\mathbb{E}X_k)^2 = o(1)$, which is exactly the estimate that makes Chebyshev's inequality deliver probability tending to one that a tree of the smaller critical size exists.

Reading between the lines

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

The exponent $(e-2)/(3e-2)$ is likely an artifact of the last interval's ratio test rather than a natural phase transition; the paper says so itself, and replacing the crude ratio lower bound by a sharper estimate could push the range further down.
Because the Stirling estimates in Section 2 assume $k=o(\sqrt{n})$ and $k\approx 2\ln(np)/p$, the present method cannot reach $p\le n^{-1/2}$; any extension below that would need a different counting argument, not just sharper constants.
The same overlap-counting machinery should transfer to maximum induced forests: the difficult near-complete-overlap range is a purely combinatorial statement about extending a fixed forest to a tree, so the improved bounds would apply there as well.
One testable way to calibrate the theorem is to simulate $G(n,p)$ at, say, $p=n^{-1/10}$ and compare the observed maximum induced tree to $2\log_{1/(1-p)}(enp)$; this would estimate the additive constant $\delta$ and check whether finite-n deviations ever exceed one.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I read the Buitrago Oropeza paper. The result is real: it moves two-point concentration for the maximum induced tree from constant p down to p as small as n^{-(e-2)/(3e-2)+epsilon}, and the proof is mostly careful. The main thing to check in review is the reliance on bounds from Kamaldinov-Skorkin-Zhukovskii [21] for f(k,l,r), the number of k-trees extending a given forest on l vertices. The paper asserts these bounds hold for every p in (0,1), but it does not reproduce the proofs, and they are load-bearing in the variance estimate (11). In the critical range l >= k - O((ln k)/p), the factor ((1-p)/p)^r is exponentially large in 1/p, so if the [21] bounds had any hidden restriction to constant p or to l not too close to k, the argument would collapse. That is a genuine concern, but it is an addressable one: a referee should ask the author to state exactly which inequalities are imported and to verify they cover the range l = k - beta/p with beta bounded.

What the paper does well is the treatment of the overlap range where the common vertices are almost all of the tree. That is not a trivial adaptation of the constant-p proof; it needs new estimates for the maximum of H(k,l,r), and the four-interval partition of l is well chosen. The threshold k-hat is computed explicitly from the expectation, and the existential delta in Theorem 1.7 is not fitted to make the proof work - it comes from the asymptotic expansion. I also appreciate that the author openly says the threshold is probably not optimal.

The soft spots, besides the [21] import, are minor. The alignment between the theorem's g(n) and the proof's k-hat is plausible from equation (4) but not spelled out clearly; a short paragraph explaining that delta absorbs the constants would fix it. The long asymptotic estimates in Section 3 are hard to check line by line, but I did not find an internal contradiction.

This paper is for specialists in random graphs and combinatorial concentration. It deserves a serious referee, not a desk rejection. The reader's CONDITIONAL verdict is about right; the issues are presentation and verification of imported bounds, not a demonstrated error. I would send it to review and ask for a careful check of the [21] imports, or for a short appendix proving the needed bounds in the sparse range.

Referee Report

2 major / 4 minor

Summary. The paper proves a two-point concentration theorem for the maximum size of an induced subtree in the binomial random graph G(n,p) in the moderately sparse regime n^{-(e-2)/(3e-2)+ε} ≤ p = o(1). Following the framework of Kamaldinov, Skorkin, and Zhukovskii, the author computes the threshold k̂ from the first moment, bounds the variance of the number X_k of induced subtrees via a decomposition over the overlap size ℓ of pairs of k-vertex trees, and partitions the possible ℓ into four ranges in the main sparse case p < 1/(2 ln n). A separate adaptation of the constant-p argument is given for p ≥ 1/(2 ln n). The main result is Theorem 1.7, asserting that for every ε>0 there is a constant δ>0 such that the maximum induced-tree size is asymptotically almost surely either g(n) or g(n)+1, with g(n)=⌊2 log_{1/(1-p)}(enp)+δ⌋.

Significance. If the theorem is correct, it is a genuine extension of the constant-p two-point concentration result of Kamaldinov, Skorkin, and Zhukovskii to a broad sparse range, improving earlier O(1/p) estimates for the maximum induced tree to a sharp two-point statement. The paper is careful about the main variance decomposition and identifies the correct threshold from the first moment, and the four-interval split of the overlap parameter is a sensible strategy for handling the sparse regime. The author also honestly notes that the lower bound on p is likely not optimal. However, the proof relies on a long chain of asymptotic estimates, two of which need attention: the imported counting bounds from [21] are used in a parameter range where their validity is not demonstrated in the manuscript, and the maximization of the auxiliary function f_1 in Section 4 appears to omit a boundary case and to undercount the maximum in (39). These issues are substantial enough that the paper needs revision before the theorem can be considered established, although the gaps appear repairable.

major comments (2)

[Section 4, Eqs. (38)–(41)] The maximization of f_1(k,r) is incomplete and the value substituted in (39) is not a valid upper bound as written. Setting ℓ=k-s and r=ℓ-x, one has ln f_1 = (s+x) ln(ℓA/x) with A=ps/(1-p). The stationary point is x_* = ℓA/e. In interval 1, where s ≥ 9(1-p)/(8p), one has A>1 and hence x_* > ℓ/e, so on the branch r ≥ ℓ(1-1/e) the function is monotone in r and its maximum on that branch occurs at the boundary r=ℓ(1-1/e), not at r ∼ ℓ(1-ps/((1-p)e)). The same happens in the upper part of interval 2. Moreover, the quantity ℓps/((1-p)e) - s used in (39) is smaller than the actual boundary value by roughly 2s + s ln A plus lower-order terms, so the displayed equality replacing the maximum by that expression is not justified. The later cases (40)-(41) do not cover the boundary maximum. Since this analysis underpins the entire p ≥ 1/(2 ln n) regime, the proof needs to be repaired, for example by showing that the boundary case is bounded by a constant multiple of one of the cases already treated.
[Sections 3.2–3.4] The proof imports the bounds on f(k,ℓ,r) from [21] and uses them in the sparse regime in a load-bearing way. In Section 3.2 the paper states that f(k,ℓ,r)((1-p)/p)^r ≤ (k-ℓ)^{k-2}(ℓ+1)^{k-ℓ-1} for all p∈(0,1), and Sections 3.3.1-3.3.3 use the piecewise formulas (19), (24), and (25) without reproducing their derivation or giving a numbered lemma. The range in which these bounds are applied includes k-ℓ = β/p with β as small as a constant, where ((1-p)/p)^r is exponentially large in 1/p; a hidden condition on p or on k-ℓ would invalidate the estimates leading to (17), (31), and (34). The author should state the exact lemma from [21] with all hypotheses, or include a proof in an appendix, so that a reader can verify that no constant-p condition is being imported into the sparse setting.

minor comments (4)

[Section 2.1, Eq. (4)] The correction term 3 ln p/(2 ln(np)) in the formula for k̂ is not o(1) in general (for p=n^{-α} it is a bounded nonzero constant depending on α), so the paper should explain explicitly how this bounded correction is absorbed by the constant δ in Theorem 1.7 and how the thresholds in Section 2.2 correspond to the two values g(n) and g(n)+1.
[Sections 3.3.1-3.3.3] The notation in the displayed formulas for f(k,ℓ,r), such as '32r−ℓ22ℓ−3r', is ambiguous and should be written as 3^{2r-ℓ}2^{2ℓ-3r}; similarly, the expression in (24) should be checked for typographical consistency with the surrounding text.
[Section 4] There are several typographical slips: 'dicreases' for 'decreases', 'asypmtotics' for 'asymptotics', and the phrase 'in interval 2' appears where 'interval 1' is clearly intended in the paragraph following (39).
[Section 1, Theorem 1.7] The theorem statement does not make explicit whether δ may depend on ε; the proof requires this, and the statement should say 'for every ε>0 there exists a constant δ=δ(ε)>0'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof is self-contained given the external bounds from [21]; the only self-citation is a preliminary announcement and is not load-bearing.

full rationale

The central claim, two-point concentration of the maximum induced tree for n^{-(e-2)/(3e-2)+eps} <= p = o(1), is derived from first and second moment estimates on the number X_k of induced k-trees, not from the theorem being used as an input. The value k̂ is obtained by solving gamma(k)=0, where gamma is the log of EX_k; equation (4) then expresses k̂ explicitly as 2 log_{1/(1-p)}(enp) plus lower-order terms. The constant delta in Theorem 1.7 is existential and is shown to exist from this explicit expansion, so it is not a fitted parameter renamed as a prediction. The proof relies on the external bound for f(k,ell,r) from Kamaldinov, Skorkin and Zhukovskii [21], and on the rooted-forest count from Moon [25]; these are independent external results, not self-citations, and they do not contain the target conclusion. The only self-citation, [8], is merely the note 'A short preliminary version of this paper appeared without proofs in [8]' and is not load-bearing. The skeptic concern that the f-bounds might fail in the sparse range is a correctness risk about an imported external estimate, not circularity: the paper does not define or fit those bounds in terms of the quantity being predicted. The unstated alignment between k̂ and g(n) is also a presentational gap, not a circular reduction. The manuscript even flags its own suspected suboptimality of the p-range, which supports the interpretation that the result is a genuine extension rather than a repackaged input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard counting formulas (Cayley, Stirling, Moon) and on the bounds from [21]. No new particles or entities are introduced. The only hand-chosen constants are δ and the auxiliary sequence w(n).

free parameters (3)

δ = not specified (existential)
The constant δ in Theorem 1.7 is asserted to exist but its value is not given. The proof shows that a suitable δ can be chosen based on the asymptotic location of k̂, but the alignment is not made explicit.
w(n) = any sequence with w→∞ and w=o(√ln n)
An auxiliary sequence used in Sections 3.2 and 3.3 to split the overlap range. The proof assumes such a sequence exists but does not specify it.
ε = positive constant, input to the theorem
The ε in the lower bound on p is an assumption of the theorem, not fitted to data. It is listed because it is a hand-chosen parameter that appears in the final threshold.

assumptions (5)

standard math Cayley's formula: the number of labeled trees on k vertices is k^{k-2}
Used in equation (2) for the expectation of X_k.
standard math Stirling's approximation for factorials
Used to derive the asymptotic form of binomial coefficients and the expression for γ(k).
standard math Moon's formula for rooted forests: the number of rooted forests on n vertices with m trees is (n choose m) m n^{n-m+1}
Cited from [25] and used in Section 3.3 to bound φ(ℓ,r), the number of forests on ℓ vertices with r edges.
domain assumption The bound f(k,ℓ,r)((1-p)/p)^r ≤ (k-ℓ)^{k-2}(ℓ+1)^{k-ℓ-1} for all r, ℓ ≤ k-2(1-p)/p, all p∈(0,1), proved in [21]
Imported from Kamaldinov et al. and used in Sections 3.2 and 3.4 to bound the number of trees extending a given forest.
domain assumption The piecewise formulas for f(k,ℓ,r) on r∈[0,ℓ/2), [ℓ/2,ℓ(1-1/e)), and [ℓ(1-1/e),ℓ-1) from [21]
Used in Section 3.3 to compute the maximum of H(k,ℓ,r) over r; the formulas are stated as quoted from [21].

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Cite this review

Pith. "Pith review of Concentration of the maximum size of an induced subtree in moderately sparse random graphs." pith.science (2026). https://pith.science/paper/P6SQNPPO

@misc{pith2026250602801,
  author       = {Pith},
  title        = {Pith review of: Concentration of the maximum size of an induced subtree in moderately sparse random graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P6SQNPPO}},
  note         = {Machine review of arXiv:2506.02801}
}

abstract

Kamaldinov, Skorkin, and Zhukovskii proved that the maximum size of an induced subtree in the binomial random graph $G(n,p)$ is concentrated at two consecutive points, whenever $p\in(0,1)$ is a constant. Using improved bounds on the second moment of the number of induced subtrees, we show that the same result holds when $n^{-\frac{e-2}{3e-2}+\varepsilon}\leq p=o(1)$.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [21]

Maximum sparse induced subgraphs of the binomial random graph with given number of edges

D. Kamaldinov, A. Skorkin, M. Zhukovskii, “Maximum sparse induced subgraphs of the binomial random graph with given number of edges”,Discrete Mathematics, 2021; 344(2): 112675. 26

work page 2021

[2] [1]

The maximum size of an induced forest in the binomial random graph

M. Akhmejanova, V. Kozhevnikov, “The maximum size of an induced forest in the binomial random graph”, Discrete Mathematics, 2024; 347(7):114030. 25

work page 2024

[3] [2]

Maximum induced trees and forests of bounded degree in random graphs

M. Akhmejanova, V. Kozhevnikov, M. Zhukovskii, “Maximum induced trees and forests of bounded degree in random graphs”, 2024 preprint, arXiv:2408.15215

work page Pith review arXiv 2024

[4] [3]

On the sizes of large subgraphs of the binomial random graph

J. Balogh, M. Zhukovskii, “On the sizes of large subgraphs of the binomial random graph”, Discrete Mathematics, 2022; 345(2): 112675

work page 2022

[5] [4]

Two-Point Concentration of the Independence Number of the Ran- dom Graph

T. Bohman, J. Hofstad, “Two-Point Concentration of the Independence Number of the Ran- dom Graph”, Forum of Mathematics, Sigma, 2022; 12

work page 2022

[6] [5]

Two-point concentration of the domination number of random graphs

T. Bohman, L. Warnke, E. Zhu, “Two-point concentration of the domination number of random graphs”, 2024

work page 2024

[7] [6]

Random Graphs

B. Bollobás, “Random Graphs”, 2nd ed., Cambridge University Press, 2001

work page 2001

[8] [7]

Cliques in random graphs

B. Bollobás, P. Erdős, “Cliques in random graphs”, Math. Proc. Camb. Phil. Soc., 1976; 80: 419–427

work page 1976