Pith. sign in

REVIEW 1 major objections 4 minor 1 cited by

Disjoint pairs in set systems and combinatorics of low rank matrices

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves the optimal disjoint-pairs bound for large set families, resolving the 40-year-old problem, and extends the same covering and discrepancy machinery to dense disjointness graphs, nonzero intersections, and low-rank…

desk verdict This is a strong paper that settles the Daykin–Erdős problem with optimal dependence and proves the Singer–Sudan conjecture; the main proofs hold up, but Section 3.1 contains a false inequality and Theorem 1.10 has a fixable small-family gap. read the letter →

arxiv 2411.13510 v1 pith:FDTBICA4 submitted 2024-11-20 math.CO

Zach Hunter , Aleksa Milojević , Benny Sudakov , István Tomon This is my paper · ORCID
classification math.CO MSC 05D0505D4015A0368Q17
keywords Daykin–Erdősproblemdisjointpairssetsystemslow-rankmatriceslog-rankconjectureconstantsubmatrixcover-freefamiliesentropymethod
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 settles a 40-year-old extremal problem about set systems: how many disjoint pairs of sets can a family of large size contain? The authors prove that a family with $m = 2^{(1/2+\delta)n}$ sets on an $n$-element ground set has at most $m^2 2^{-c\delta/\log(1/\delta)}$ disjoint pairs, and that a construction proposed in 1985 reaches this bound up to the constant $c$. The same arguments prove a conjecture about extracting completely disjoint subfamilies from a dense disjointness graph, and give matching bounds on all-zero and constant submatrices of low-rank matrices, with direct consequences for the log-rank conjecture and for cover-free codes. The proofs run through a covering argument using 'bad' sets, an entropy lemma that controls the growth of random unions, and a discrepancy bound for low-rank matrices.

What carries the argument

Three mechanisms carry the proof. The first is the bad-set covering argument: a set $U$ is bad for a family $\mathcal F$ if it covers fewer than $2^{-2n/k}|\mathcal F|$ members; the union of $k$ independent random sets is bad with probability at most $2^{-n}$, while the expected common neighbourhood of $k$ random vertices in a $\delta$-dense disjointness graph is at least $\delta^k|\mathcal B|$, so a typical union yields large cross-disjoint subfamilies. The second is the entropy covering lemma: for every $\mathcal F$ of size $2^{\alpha n}$, independent uniform draws $F_1,\dots,F_k$ satisfy $\mathbb E|\bigcup_i F_i|\ge \alpha n - Cn/2^k$, proved by writing the expected union as $\sum_i(1-(1-p_i)^k)$ and comparing each term with the binary entropy $H(p_i)$; this makes random unions provably large and creates the contradiction with a small common neighbourhood. The third is a discrepancy bound for low-rank matrices: representing $M$ through singular vectors scaled by singular values and applying Grothendieck's inequality gives a half-size submatrix $M'$ with $\|M\|_C \ge c\sqrt{mn}\,\|M'\|_F^2/(\sqrt r\|M\|_F)$, where $\|\cdot\|_C$ is the cut norm; iterating this discrepancy while reducing the average entry produces large all-zero or constant submatrices.

What would settle it

Exhibit a family $F\subseteq 2^{[n]}$ of size $2^{(1/2+\delta)n}$ whose disjoint-pair density is $2^{-o(\delta/\log(1/\delta))}$, which would beat the theorem's upper bound for every constant $c$; alternatively, find a family for which the expected union of $k$ uniform random members is smaller than $\alpha n - Cn/2^k$, which would refute the lemma that drives the proof.

Watch

Extended reading notes

Core claim

The central claim is that the density of disjoint pairs in a family is controlled by the exponent $\delta$ through the ratio $\delta/\log(1/\delta)$. For $F\subseteq 2^{[n]}$ with $|F|=m\ge 2^{(1/2+\delta)n}$, the number of disjoint unordered pairs is at most $m^2 2^{-c\delta/\log(1/\delta)}$, and the construction from [2] with a split ground set and low-intersection layers has size $2^{(1/2+\delta)n}$ and contains $m^2 2^{-O(\delta/\log(1/\delta))}$ disjoint pairs, so the upper bound is optimal up to the constant $c$. In the bipartite form, two families of size at least $c_d\,2^{n/2} n^d$ have at most $(1+o(1))2^{-2d}|A||B|$ disjoint pairs. The same covering mechanism proves that families with a constant density of pairs having intersection exactly $\lambda\neq 0$ still have size at most $2^{(1/2+o(1))n}$, and that disjointness density $\delta$ forces cross-disjoint subfamilies of product size $|A||B|2^{-O(\sqrt{n\log(1/\delta)})}$. For matrices, the paper shows that an $m\times n$ rank-$r$ matrix with entries $0$ or at least $1$ and average $\varepsilon\le 1/2$ contains an all-zero submatrix of size $n2^{-O(\log r+\sqrt{\varepsilon r})}$, optimal when $\varepsilon\gg(\log r)^2/r$, and that integer matrices with entries $0,\dots,t$ contain constant submatrices of size $2^{-O(t\sqrt r)}m\times 2^{-O(t\sqrt r)}n$.

Load-bearing premise

The load-bearing premise is the entropy covering lemma: for every family $\mathcal F$ of size $2^{\alpha n}$, the union of $k$ independent uniform random sets from $\mathcal F$ has expected size at least $\alpha n - Cn/2^k$ with an absolute constant $C$; if that universal union-growth bound failed, the dependent-random-choice contradiction behind Theorems 1.2 and 1.3 would collapse.

Editorial extensions

If this is right

  • The 1985 construction is optimal up to a constant factor, so the maximum disjoint-pair density for families of size $2^{(1/2+\delta)n}$ is $2^{-\Theta(\delta/\log(1/\delta))}$.
  • The biclique conjecture of [26] holds in strong quantitative form: disjointness density $\delta$ always yields cross-disjoint subfamilies of size at least $|A||B|2^{-O(\sqrt{n\log(1/\delta)})}$.
  • For every distribution $\mu$ on $2^{[n]}$, the probability that $A_0\subseteq A_1\cup\cdots\cup A_r$ for independent draws is at least $2^{-n/r-2}$, which is tight up to the constant and sharpens the known supersaturation bound for $r$-cover-free families.
  • The sparse low-rank matrix conjecture from [22] is confirmed for separated matrices: if the average entry of a rank-$r$ matrix is $\varepsilon\le1/2$, an all-zero submatrix of size $n2^{-O(\log r+\sqrt{\varepsilon r})}$ exists, and this is optimal in the regime $\varepsilon\gg(\log r)^2/r$.
  • Log-rank-type rectangle bounds extend beyond binary matrices: integer matrices with entries $0,\dots,t$ and rank $r$ contain a constant submatrix of size $2^{-O(t\sqrt r)}m\times 2^{-O(t\sqrt r)}n$.

Reading between the lines

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

  • The entropy covering lemma suggests a transferable principle: for any large family, $k$ uniform random draws cover essentially as many coordinates as $k$ draws from a subcube of dimension $\log_2|\mathcal F|$. If this principle extends to other forbidden intersection patterns, the $\lambda\neq0$ variant of the disjoint-pairs problem could be sharpened from constant-density to subconstant density.
  • Theorem 1.12 shows the additive $\log r$ loss in the all-zero rectangle bound is only needed when the average entry is very small; reaching the conjectured $2^{-O(\sqrt{\varepsilon r})}$ for all $\varepsilon$ would require a new argument near the boundary $\varepsilon\sim(\log r)^2/r$, which the construction in the paper shows is the delicate regime.
  • The bad-set covering proof of Lemma 1.11 is distribution-free and yields a quantitative covering probability; the same union-bound-over-bad-sets trick should give supersaturation bounds for other union-closed properties, such as $k$-wise disjointness or prescribed intersection sizes, with the same $2^{-O(n/r)}$ shape.
  • The bipartite reduction in the proof of Theorem 1.3 indicates that sharp constants for other intersection constraints might follow from the same discrepancy and entropy template; Problem 6.2 of the paper, whether a large family of pairs with intersection $\lambda$ contains a large subfamily with all intersections exactly $\lambda$, is a natural first test.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper studies extremal problems about disjoint pairs in set systems and zero/constant submatrices in low-rank matrices. Its main results are: an optimal bound for the Daykin-Erdős problem with the conjectured dependence on δ (Theorem 1.2); a bipartite strengthening with asymptotically sharp constants (Theorem 1.3); a Singer–Sudan type biclique theorem for disjointness graphs (Theorem 1.10); a tight lower bound for r-cover-free families (Lemma 1.11); a nonzero-intersection analogue (Theorem 1.5); an even/odd intersection bound (Theorem 1.6); new low-rank matrix results for sparse separated matrices (Theorem 1.12) and for integer matrices with bounded average (Theorem 1.13). The proofs combine dependent random choice, entropy, Fourier analysis, discrepancy, and additive combinatorics, and the paper also provides constructions showing optimality of several bounds.

Significance. If the proofs are correct, these are substantial contributions. Theorem 1.2 resolves the Daykin-Erdős problem with the optimal dependence on δ, matching Construction 1 up to the constant c. Theorem 1.10 proves the Singer–Sudan conjecture in a strong quantitative form, and Lemma 1.11 settles the Alon–Gilboa–Gueron problem up to a constant in the exponent. The matrix results generalize the best known log-rank bounds and are optimal in a natural parameter range. The entropy proof of Theorem 1.2 via Theorem 3.3 and Lemma 3.5 is sound; I checked the dependent-random-choice chain and the constants. The combinatorial proof of Theorem 3.2 is also essentially correct once the notation '104' is read as 10^4, matching the hypothesis of Theorem 3.2; with that reading the numerical inequalities are valid. The main weakness is in the appendix proof of Theorem 1.13, which has a missing verification of a key hypothesis.

major comments (1)
  1. [Appendix, proof of Theorem 1.13] In the key statement of the proof, Proposition A.5 is applied to the matrix M − ℓJ, but the hypothesis of Proposition A.5 that all entries lie in (−r^3, r^3) is not verified. Lemma 4.7(1) only shows that a submatrix has entries bounded by 400r^2 p(M), which can exceed r^3 when ℓ grows, and M − ℓJ can have negative entries of similarly large magnitude. This matters because the proof of Proposition A.5 uses the entry bound to assert q0 ≤ r^6 and hence f(p0) + g(q0) = O(√r + log r); without that bound the potential could be much larger and the claimed O(√r) iteration count does not follow. The theorem may still be true, and the gap may be repairable by a capping argument or by splitting off the regime where t is large, but as written the proof of Theorem 1.13 is incomplete.
minor comments (4)
  1. [Section 3.1, proof of Theorem 3.2] The apparent numerical contradiction in the displayed inequality disappears when '104' is read as 10^4, as in the theorem statement; with that reading 32/100 + 192/10^4 < 1/2 and the analogous final bound 35/100 + 210/10^4 < 1 also hold. The text should use a clear superscript to avoid confusion.
  2. [Section 4, Claim 4.1] The proof of the first bullet of Claim 4.1 is omitted with a pointer to the authors' own paper [28]. Since this claim is used in the proof of Theorem 1.12, the manuscript should either include the short proof or state the precise claim from [28] being invoked.
  3. [Statement of Theorem 1.12] The theorem is stated for an m × n matrix but the conclusion says 'an all-zero submatrix of size at least n2^{-O(log r+√εr)}'. If m is much smaller than n this is false as stated; the proof actually gives a submatrix of size 2^{-O(...)}m × 2^{-O(...)}n. The statement should be corrected to use min(m,n) or to state both dimensions.
  4. [Section 2, proof of Theorem 1.10] The small-family case is written only for |B| ≤ 2^{2√{n log δ^{-1}}}; if instead |A| is small while |B| is large, the symmetric argument picking one element of A and its neighbourhood in B should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main theorems are proved from first principles against external benchmarks and constructions.

full rationale

The central results do not reduce to their inputs by construction. Theorem 1.2 is proved twice: the combinatorial proof in Section 3.1 is self-contained given Lemma 3.1, which is proved in the text; the entropy proof in Section 3.2 rests on Lemmas 3.4 and 3.5, both proved in-text using only elementary inequalities and the standard subadditivity of entropy (cited to Alon–Spencer). Theorem 1.3 follows from Theorem 3.3, whose proof is again self-contained; the Alon–Frankl theorem is used only in the reduction to the bipartite setting, not as an input to the main bound. Theorem 1.5 uses the external results of Gowers–Green–Manners–Tao, Ben-Sasson–Lovett–Ron-Zewi, Bhowmick–Dvir–Lovett, and Sgall; none of these assume the paper's conclusions. Theorems 1.12 and 1.13 are proved via discrepancy lemmas developed in Section 4 and the Appendix; Lemma 4.2 is credited to Kwan and Sauermann, and the iteration arguments are proved in-text. The only deferred item is Claim 4.1, whose proof is said to be 'almost identical' to Claim 2.2 of the authors' own [28]; this is a parameter-free technical averaging fact about discrepancy and does not assume any of the target theorems, so the citation is real evidence rather than a load-bearing self-citation. The lower bounds in Constructions 1, 2, and 3 serve as external benchmarks, and the upper bounds are matched against them rather than derived from them. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' prior work is invoked to force a choice, and no known result is merely renamed. Hence the derivation chain is self-contained apart from minor, non-circular deferrals.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper rests on several established external theorems: Grothendieck's inequality and the cut-norm/γ_2^* equivalence, the partial Approximate Duality result of Ben-Sasson-Lovett-Ron-Zewi and Bhowmick-Dvir-Lovett (now unconditional via the PFR theorem of Gowers-Green-Manners-Tao), Sgall's bound on constant-intersection families, and the Szemerédi regularity lemma. No new postulates or fitted constants are introduced; the proofs choose universal constants that work for the stated asymptotic claims. The central claims are derived, not assumed.

assumptions (6)
  • standard math Grothendieck's inequality: the cut norm and the γ_2^* norm of a real matrix are equivalent up to an absolute constant.
    Used in Lemma 4.2 to convert a lower bound on the γ_2^* norm (from an SVD construction) into a lower bound on the cut norm, which is then used to derive discrepancy lower bounds for low-rank matrices.
  • standard math Approximate Duality Theorem (Theorem 3.7 of the paper, from Ben-Sasson-Lovett-Ron-Zewi [5] and Bhowmick-Dvir-Lovett [7]): sets A,B in F_p^n with bias ≥ 2/(3p^{3/2}) contain large subfamilies with constant inner product.
    This is the key external result used in the proof of Theorem 1.5 (Daykin-Erdős with non-zero intersection λ). It was originally conditional on the Polynomial Freiman-Ruzsa conjecture.
  • standard math Polynomial Freiman-Ruzsa theorem over F_2^n, proven by Gowers-Green-Manners-Tao [14], used to remove the conditionality from Theorem 3.7.
    The paper's Theorem 1.5 relies on the now-unconditional status of the partial Approximate Duality result; PFR is cited as the proof.
  • standard math Sgall's bound: if A,B ⊆ 2^[r] and |A∩B| is constant over A×B, then |A||B| ≤ 2^r (Corollary 3.5 of [25]).
    Used twice: in Theorem 1.5 to convert large constant-intersection-mod-p subfamilies into a size bound, and in Lemma 5.2 to show the optimality construction M3 has no large all-zero submatrix.
  • standard math Szemerédi's regularity lemma and Mantel's theorem, used in the non-bipartite reduction from Theorem 1.3 to the original Daykin-Erdős problem.
    The reduction of the one-family statement to the bipartite Theorem 1.3 is sketched using a regularity partition and the fact that dense disjointness graphs have few triangles.
  • standard math Alon-Frankl bound [2] for disjoint pairs in families on smaller ground sets, used in the same non-bipartite reduction.
    Used to argue that the neighbourhood of a large set has o(m^2) disjoint pairs when restricted to a universe of size ≤ 9n/10.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Disjoint pairs in set systems and combinatorics of low rank matrices." pith.science (2026). https://pith.science/paper/FDTBICA4

@misc{pith2026241113510,
  author       = {Pith},
  title        = {Pith review of: Disjoint pairs in set systems and combinatorics of low rank matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDTBICA4}},
  note         = {Machine review of arXiv:2411.13510}
}
abstract

We study and solve several problems in two closely related settings: set families in $2^{[n]}$ with many disjoint pairs of sets and low rank matrices with many zero entries. - More than 40 years ago, Daykin and Erd\H{o}s asked for the maximum number of disjoint pairs of sets in a family $F\subseteq 2^{[n]}$ of size $2^{(1/2+\delta)n}$ and conjectured it contains at most $o(|F|^2)$ such pairs. This was proven by Alon and Frankl in 1985. In this paper we completely resolve this problem, proving an optimal dependence of the number of disjoint pairs on the size of family $F$. We also prove the natural variant of the Daykin-Erd\H{o}s conjecture in which disjoint pairs are replaced by pairs with intersection $\lambda\neq 0$. - Motivated by a conjecture of Lovett related to the famous log-rank conjecture, Singer and Sudan asked to show that for two families $A, B \subseteq 2^{[n]}$ with a positive constant fraction of set pairs $(a,b)\in A\times B$ being disjoint, there are $R\subset A$ and $S\subset B$ such that all set pairs $(r, s)\in R\times S$ are disjoint, and $|R|\geq 2^{-O(\sqrt{n})}|A|$ and $|S|\geq 2^{-O(\sqrt{n})}|B|$. We prove this conjecture in a strong quantitative form. - We prove the following generalizations of the best known bounds for the log-rank conjecture. If $M$ is an $n\times n$ non-negative integer matrix of rank $r$ in which the average of the entries is $\varepsilon\leq 1/2$, then $M$ contains an all-zero submatrix of size at least $2^{-O(\sqrt{\varepsilon r})}n$. Unlike the known bounds for the log-rank conjecture, this result is optimal. Moreover, using similar methods, we also prove that any $n\times n$ matrix of rank $r$ with entries from $\{0,\dots,t\}$ contains a constant submatrix of size at least $2^{-O(t\sqrt{r})}n$. Our proofs use probabilistic, entropy and discrepancy methods and explore connections to additive combinatorics and coding theory.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Factorization norms and an inverse theorem for MaxCut

    math.CO 2025-06 conditional novelty 8.0 of 10

    Boolean matrices with bounded γ2-norm or normalized trace norm contain linear-sized homogeneous submatrices, and graphs with near-minimal MaxCut contain a large clique.

Reference graph

Works this paper leans on

28 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [28]

    Sudakov and I

    B. Sudakov and I. Tomon. Matrix discrepancy and the log-r ank conjecture. Mathematical Programming, 2024. 23 A Appendix In this appendix, we give a proof of our final result mentioned in the Introduction, namely that matrices with small integer entries also have large monochromatic re ctangles. The motivation for this theorem, as for many others in this pa...

    work page 2024

  2. [18]

    Hunter, A

    Z. Hunter, A. Milojević, B. Sudakov, and I. Tomon. Kővári -Sós-Turán theorem for hereditary families. preprint, arXiv:2401.10853

    arXiv

  3. [23]

    Milojević, B

    A. Milojević, B. Sudakov, I. Tomon. Incidence bounds via extremal graph theory. preprint, arxiv:2401.06670

    arXiv

  4. [1]

    N. Alon, S. Das, R. Glebov, and B. Sudakov. Comparable pair s in families of sets. Journal of Combinatorial Theory, Series B , 115:164–185, 2015

    work page 2015

  5. [2]

    Alon and P

    N. Alon and P. Frankl. The maximum number of disjoint pair s in a family of subsets. Graphs Combin., 1(1):13–21, 1985

    work page 1985

  6. [3]

    N. Alon, S. Gilboa, and S. Gueron. A probabilistic varian t of Sperner’s theorem and of maximal r-cover free families. Discrete Math. , 343(10):112027, 4, 2020

    work page 2020

  7. [4]

    Alon and J

    N. Alon and J. H. Spencer. The Probabilistic Method. John Wiley & Sons, 2016

    work page 2016

  8. [5]

    Ben-Sasson, S

    E. Ben-Sasson, S. Lovett, and N. Ron-Zewi. An additive com binatorics approach relating rank to communication complexity. J. ACM , 61(4), July 2014

    work page 2014

Show all 28 references
  1. [6]

    Ben-Sasson and N

    E. Ben-Sasson and N. Ron-Zewi. From affine to two-source ext ractors via approximate duality. SIAM Journal on Computing , 44(6):1670–1697, 2015

    2015

  2. [7]

    Bhowmick, Z

    A. Bhowmick, Z. Dvir, and S. Lovett. New bounds for matchin g vector families. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing , pages 823–832, New York, NY, USA, 2013. Association for Computing Machinery

    2013

  3. [8]

    Chazelle

    B. Chazelle. Cutting hyperplanes for divide-and-conque r. Discrete & Computational Geometry , 9(2):145–158, 1993

    1993

  4. [9]

    A. G. D‘yachkov and V. V. Rykov. Bounds on the length of disj unctive codes. Problems Inform. Transmission, 18(3):166–171, 1982

    1982

  5. [10]

    Erdős, P

    P. Erdős, P. Frankl, and Z. Füredi. Families of finite set s in which no set is covered by the union of r others. Israel J. Math. , 51(1-2):79–89, 1985

    1985

  6. [11]

    Erdős, C

    P. Erdős, C. Ko, and R. Rado. Intersection theorems for s ystems of finite sets. Quart. J. Math. Oxford Ser.(2) , 12:313–320, 1961

    1961

  7. [12]

    Fox and B

    J. Fox and B. Sudakov, Density theorems for bipartite gra phs and related Ramsey-type results, Combinatorica 29 (2009) 153-196

    2009

  8. [13]

    Z. Füredi. On r-cover-free families. J. Combin. Theory Ser. A , 73(1):172–173, 1996

    1996

  9. [14]

    W. T. Gowers, B. Green, F. Manners, and T. Tao. On a conject ure of Marton. preprint, arXiv:2311.05762

    arXiv

  10. [15]

    W. T. Gowers, B. Green, F. Manners, and T. Tao. Marton’s co njecture in abelian groups with bounded torsion. preprint, arXiv:2404.02244. 22

    arXiv

  11. [16]

    Grothendieck

    A. Grothendieck. Résumé de la théorie métrique des prod uits tensoriels topologiques, volume 2. Soc. de Matemática de São Paulo , 1956

    1956

  12. [17]

    R. Guy. Unsolved Problems: A Miscellany of Erdős Proble ms. Amer. Math. Monthly , 90(2):118+119–120, 1983

    1983

  13. [19]

    Kautz and R

    W. Kautz and R. Singleton. Nonrandom binary superimpos ed codes. IEEE Transactions on Information Theory , 10(4):363–377, 1964

    1964

  14. [20]

    Linial and A

    N. Linial and A. Shraibman. Learning complexity vs comm unication complexity. Combinatorics, Probability and Computing , 18(1-2):227–245, 2009

    2009

  15. [21]

    Lovász and M

    L. Lovász and M. Saks. Communication complexity and com binatorial lattice theory. volume 47, pages 322–349. 1993. 29th Annual IEEE Symposium on Foundati ons of Computer Science (White Plains, NY, 1988)

    1993

  16. [22]

    S. Lovett. Communication is bounded by root of rank. J. ACM , 63(1):Art. 1, 9, 2016

    2016

  17. [24]

    I. Rival. In Ordered sets : proceedings of the NATO Advanced Study Instit ute held at Banff, Canada, August 28 to September 12, 1981 , NATO advanced study institutes series. Series C, Mathemat ical and physical sciences ; Volume 83, page 860, Dordrecht, The N etherlands ;, 1982...

    1981

  18. [25]

    J. Sgall. Bounds on pairs of families with restricted int ersections. Combinatorica, 19(4):555–566, 1999

    1999

  19. [26]

    Singer and M

    N. Singer and M. Sudan. Point-hyperplane incidence geo metry and the log-rank conjecture. ACM Trans. Comput. Theory , 14(2):Art. 7, 16, 2022

    2022

  20. [27]

    Smorodinsky

    S. Smorodinsky. A survey of Zarankiewicz problem in geo metry. preprint, arXiv:2410.03702

    arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.