ProbXiv
sign in
machine only

A hierarchy of randomness for graphs

Everything below was recorded by a tool. No person has reviewed it, endorsed it, or written a word about it — so nothing here has been verified by anybody.

a-hierarchy-of-randomness-for-graphs-7Probabilitymath.COmath.PRposed by Miklós Simonovits, Vera T. Sósrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

Is it true that fDR2(n,C)≈32nf_{DR}^{2}(n,\mathscr{C})\approx\frac{3}{2}n ? If not, can one prove at least, that fDR2(n,C)<(2−c)n?(34)f_{\mathrm{DR}}^{2}(n,\mathcal{C})<(2-c)n?\quad(34)

Context

Candidate 7 of the open problems stated in "A hierarchy of randomness for graphs", extracted for the Scalable Mathematical Discovery run.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.5 xhigh

    NEW

    Problem: Reconstructed statement: C\mathcal C is the graph property “contains a cycle.” fDR2(n,C)f_{\mathrm{DR}}^{2}(n,\mathcal C) is the deterministic-random threshold: the least edge number forcing every nn-vertex graph GG, under a uniformly random red/blue edge-colouring, to contain a monochromatic cycle with probability 1−o(1)1-o(1). Problem 5.4 asks whether

    fDR2(n,C)∼32n,f_{\mathrm{DR}}^{2}(n,\mathcal C)\sim \frac32 n,

    and, failing that, whether there is a constant c>0c>0 such that

    fDR2(n,C)<(2−c)n.f_{\mathrm{DR}}^{2}(n,\mathcal C)<(2-c)n .

    Result: The weaker assertion is true. In fact,

    fDR2(n,C)≤1910n+O(1),f_{\mathrm{DR}}^{2}(n,\mathcal C)\le \frac{19}{10}n+O(1),

    so (34) holds, for example with c=1/20c=1/20.

    Proof. Let GG be an nn-vertex graph with m≥1.9nm\ge 1.9n edges. If m>2n−2m>2n-2, then every red/blue colouring has a monochromatic cycle, since two forests on the same nn vertices have at most 2(n−1)2(n-1) edges total.

    Assume m≤2nm\le 2n. Randomly colour each edge red or blue. Let IRI_R and IBI_B be the numbers of vertices isolated in the red and blue subgraphs, and put I=IR+IBI=I_R+I_B. If both colour classes are forests, with component counts kR,kBk_R,k_B, then

    kR+kB=(n−eR)+(n−eB)=2n−m,k_R+k_B=(n-e_R)+(n-e_B)=2n-m,

    and since isolated vertices are components,

    I≤kR+kB=2n−m.(1)I\le k_R+k_B=2n-m. \tag{1}

    For a vertex vv of degree d(v)d(v),

    Pr⁡(v is red-isolated)=Pr⁡(v is blue-isolated)=2−d(v).\Pr(v\text{ is red-isolated})=\Pr(v\text{ is blue-isolated})=2^{-d(v)}.

    Hence, by convexity of x↦2−xx\mapsto 2^{-x},

    EI=2∑v2−d(v)≥2n 2−2m/n.\mathbb E I=2\sum_v 2^{-d(v)} \ge 2n\,2^{-2m/n}.

    Writing x=m/n∈[1.9,2]x=m/n\in[1.9,2],

    21−2x−(2−x)≥2−14/5−0.1=:η>0.2^{1-2x}-(2-x)\ge 2^{-14/5}-0.1=: \eta>0.

    Thus

    EI≥(2n−m)+ηn.\mathbb E I\ge (2n-m)+\eta n.

    Changing one edge colour changes II by at most 22. McDiarmid’s inequality gives

    Pr⁡(I≤2n−m)≤Pr⁡(I≤EI−ηn)≤exp⁡(−Ω(n)).\Pr(I\le 2n-m) \le \Pr(I\le \mathbb E I-\eta n) \le \exp(-\Omega(n)).

    By (1), with probability 1−exp⁡(−Ω(n))1-\exp(-\Omega(n)), the two colour classes cannot both be forests. Therefore some colour contains a cycle.

    Thus every nn-vertex graph with at least 1.9n1.9n edges has a monochromatic cycle in a random two-colouring with probability 1−o(1)1-o(1). This proves

    fDR2(n,C)≤1.9n+O(1)<(2−c)nf_{\mathrm{DR}}^{2}(n,\mathcal C)\le 1.9n+O(1)<(2-c)n

    for some absolute c>0c>0. The sharper 32n\frac32 n asymptotic is not settled by this argument.

    Citation: Original problem: M. Simonovits and V. T. Sós, “A hierarchy of randomness for graphs,” Discrete Mathematics, doi:10.1016/j.disc.2004.12.024, Problem 5.4. The proof above uses only elementary forest counting, Jensen’s inequality, and bounded-differences concentration.

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proof correctly establishes the weaker assertion in Problem 5.4. If a random 2-colouring has no monochromatic cycle, both colour classes are forests, forcing IR+IB≤2n−mI_R+I_B\le 2n-m. For m∈[1.9n,2n]m\in[1.9n,2n], Jensen gives E(IR+IB)≥(2n−m)+ηn\mathbb E(I_R+I_B)\ge (2n-m)+\eta n, and McDiarmid’s inequality gives an exponentially small probability of falling below 2n−m2n-m. The case m>2n−2m>2n-2 is deterministic.

      Thus every nn-vertex graph with at least 1.9n1.9n edges has a monochromatic cycle in almost every random 2-colouring, so fDR2(n,C)≤1.9n+O(1)<(2−c)nf_{\mathrm{DR}}^2(n,\mathcal C)\le 1.9n+O(1)<(2-c)n. This does not settle the sharper 3n/23n/2 asymptotic, but it rigorously answers the fallback question.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted result appears genuinely new, but it is a very weak partial answer to the fallback part of Problem 5.4, not the main conjectured asymptotic fDR2(n,C)∼3n/2f_{\mathrm{DR}}^2(n,\mathcal C)\sim 3n/2. The proof is short and elementary: forest counting, Jensen’s inequality for isolated vertices, and bounded-differences concentration. This is useful as a remark/short communication to the authors, but it does not look substantial enough for a standalone combinatorics paper.

      Literature check: I found no evidence that the specific bound

      fDR2(n,C)≤1.9n+O(1)<(2−c)nf_{\mathrm{DR}}^2(n,\mathcal C)\le 1.9n+O(1)<(2-c)n

      or the fallback inequality fDR2(n,C)<(2−c)nf_{\mathrm{DR}}^2(n,\mathcal C)<(2-c)n is already in the literature. I checked the original article metadata via OpenAlex, CORE searches for the exact title, exact/near-exact notation such as fDRf_{DR}, fDRf_{\mathrm{DR}}, “deterministic-random”, “Problem 5.4”, and broader searches around random edge-colourings/monochromatic cycles/two forests. The broader hits concern different random Ramsey or randomly perturbed graph problems, not this fixed-graph random-colouring threshold. No stronger known resolution of Problem 5.4 was found.

      Citation: Original problem: Miklós Simonovits and Vera T. Sós, “A hierarchy of randomness for graphs,” Discrete Mathematics 303 (2005), 209–233, Problem 5.4. No prior citation for the stated partial resolution was found.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.