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  "slug": "dependence-ultra-log-concavity",
  "title": "Ultra log-concavity for increasing families generated by singletons and pairs",
  "shortTitle": "Dependent sets of a graph are ultra log-concave",
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  "oneLine": "For any graph, the fraction of k-vertex sets that contain an edge forms a log-concave sequence in k, even with vertex weights, proving Xie and Xu's Conjecture 4.1; witnesses of size three already break it.",
  "abstract": "Pick a graph on n vertices and call a set of vertices dependent if it contains at least one edge. Let d_k count the dependent k-vertex sets. Horrocks (2002) proved that d_k is log-concave. Xie and Xu (Electronic Journal of Combinatorics, 2026) conjectured the stronger ultra log-concave form, their Conjecture 4.1: d_k/C(n,k) is log-concave, so the size of a uniformly random dependent set is at least as concentrated as a binomial distribution. This candidate proves that conjecture in a weighted form, for every increasing family generated by singletons and pairs (graphs with loops) and arbitrary nonnegative element weights: k(n−k)A_k² ≥ (k+1)(n−k+1)A_{k−1}A_{k+1}. The proof first shows that the binomially normalised number of k-sets S with both S and its complement dependent is symmetric and unimodal, a strengthening of a 2000 theorem of Horrocks, using a marked-downset lemma derived from Katona's shadow theorem and a critical-element double count; the property survives conditioning, and the Kahn–Neiman antipodal-pairs criterion converts it into ultra log-concavity. The paper also proves a weighted form of Xie and Xu's Conjecture 4.4 for additive 2-matroids of rank at most four or on at most seven elements (the general conjecture remains open), shows that witnesses of size three break the theorem (counts (0,1,4,10,5,1) give 96 < 120), and derives ultra log-concavity of noncover-complex face numbers and a variance bound for Bernoulli variables conditioned to hit an edge. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
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  "keywords": [
    "ultra log-concavity",
    "log-concavity",
    "dependent sets",
    "dependence polynomial",
    "increasing families",
    "Kahn–Neiman criterion",
    "Katona shadow theorem",
    "negative dependence",
    "2-matroids",
    "Xie–Xu conjecture"
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    "Main theorem (Theorem 1.1): for every increasing family generated by singletons and pairs and all nonnegative weights, k(n−k)A_k² ≥ (k+1)(n−k+1)A_{k−1}A_{k+1}; with unit weights this is Xie–Xu Conjecture 4.1 for the dependent-set counts d_k of every graph.",
    "Two-colouring theorem (Theorem 1.2): the number b_k of k-sets S with S and its complement both dependent, divided by C(n,k), is symmetric and unimodal; this strengthens Horrocks's 2000 unimodality theorem and survives fixing any coordinates.",
    "2-matroids (Section 6): a weighted form of Xie–Xu Conjecture 4.4 holds for additive 2-matroids of rank at most four or on at most seven elements, and the first three inequalities hold in any rank; the general conjecture remains open, with an explicit eight-element obstruction to the method.",
    "Sharpness (Section 5.3): witnesses of size three break the theorem; a five-element family with counts (0,1,4,10,5,1) gives 96 < 120, two disjoint triples with weights (1,M,M,3,3,3) fail exactly for M ≥ 44, and six elements is the minimum for a uniform-triple counterexample.",
    "Consequences (Section 5): ultra log-concave face numbers for noncover complexes, and Var(K) ≤ m(1 − m/n) for the number K of successes among independent Bernoulli variables conditioned to contain an edge, with mean m."
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  "relatedWorks": [
    {
      "citation": "Xie and Xu (2026). Ultra log-concavity and real-rootedness of dependence polynomials. The Electronic Journal of Combinatorics 33(4), P4.9. States Conjecture 4.1, proved here in weighted form, and Conjecture 4.4 on 2-matroids, proved here for rank at most four or at most seven elements.",
      "url": "https://doi.org/10.37236/13310",
      "doi": "10.37236/13310"
    },
    {
      "citation": "Horrocks (2002). The numbers of dependent k-sets in a graph are log concave. Journal of Combinatorial Theory, Series B. Proves ordinary log-concavity of the counts; its full text was not obtained for this release.",
      "url": "https://doi.org/10.1006/jctb.2001.2077",
      "doi": "10.1006/jctb.2001.2077"
    },
    {
      "citation": "Kahn and Neiman (2011). A strong log-concavity property for measures on Boolean algebras. Journal of Combinatorial Theory, Series A. The antipodal-pairs criterion used to convert the two-colouring theorem into ultra log-concavity.",
      "url": "https://doi.org/10.1016/j.jcta.2011.02.007",
      "doi": "10.1016/j.jcta.2011.02.007"
    },
    {
      "citation": "Frankl (2016). Linear independence, a unifying approach to shadow theorems. European Journal of Combinatorics. Source used for the statement of Katona's intersection-shadow theorem.",
      "url": "https://doi.org/10.1016/j.ejc.2016.04.001",
      "doi": "10.1016/j.ejc.2016.04.001"
    },
    {
      "citation": "Liggett (1997). Ultra logconcave sequences and negative dependence. Journal of Combinatorial Theory, Series A. Background on ultra log-concavity and its closure under convolution.",
      "url": "https://doi.org/10.1006/jcta.1997.2790",
      "doi": "10.1006/jcta.1997.2790"
    },
    {
      "citation": "Anari, Liu, Oveis Gharan and Vinzant (2024). Log-concave polynomials III: Mason's ultra-log-concavity conjecture for independent sets of matroids. Proceedings of the American Mathematical Society. The matroid analogue for independent sets.",
      "url": "https://doi.org/10.1090/proc/16724",
      "doi": "10.1090/proc/16724"
    },
    {
      "citation": "Brändén and Huh (2020). Lorentzian polynomials. Annals of Mathematics. A second proof of Mason's conjecture; the paper explains why this route does not apply directly to dependent sets.",
      "url": "https://doi.org/10.4007/annals.2020.192.3.4",
      "doi": "10.4007/annals.2020.192.3.4"
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