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On possible uniform Tur\'an densities

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Pith's one-line read This paper proves that every palette Lagrangian, including irrational numbers, is the uniform Turán density of some finite family of 3-graphs.

desk verdict A strong finite-family converse to Lamaison, with one local constant gap in the central stability proof that looks fixable. read the letter →

arxiv 2504.21220 v2 pith:6LQX2IOR submitted 2025-04-29 math.CO

Dylan King , Simón Piga , Marcelo Sales , Bjarne Schülke This is my paper · ORCID
classification math.CO MSC 05C6505D0505D1005D40
keywords uniformTurándensitypaletteLagrangian3-graphshypergraphproblemregularitylemmaRamseytheoryextremalcombinatorics
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 every value obtained as the Lagrangian of a finite palette is also the uniform Turán density of some finite family of 3-graphs. Palette Lagrangians are the numbers produced by the standard lower-bound constructions in this area, so this says the simplest construction scheme already realizes every such density exactly. Since palette Lagrangians include irrational numbers, it follows that finite families of 3-graphs have irrational uniform Turán densities. This complements a recent approximation result, which shows every uniform Turán density of a finite family can be approximated by palette Lagrangians; together the two results bracket the set of attainable densities between the palette Lagrangians and their closure.

What carries the argument

The load-bearing object is the palette Lagrangian $\Lambda_P$: for a palette $P$ with color set $C$, $\Lambda_P = \max\{\sum_{(i,j,k)\in P} x_i x_j x_k : x_i \ge 0,\ \sum_{i\in C} x_i = 1\}$. The main technical theorem (2.6) forces every extremal palette for a carefully chosen finite family $\mathcal{H}$ to be a blow-up of a given reduced palette $P$ or of its reverse, which pins the palette Turán density of $\mathcal{H}$ exactly at $\Lambda_P$. Three ingredients carry the proof: a palette removal lemma obtained from a regularity lemma for palettes; a structural Ramsey lemma (Lemma 4.1) that distinguishes palettes by the 3-graphs they paint, built on a Ramsey theorem for ordered Steiner systems; and a stability argument showing that palettes extremely close to a blow-up of $P$ are in fact blow-ups of $P$.

What would settle it

Compute the palette Turán density and the uniform Turán density of the finite family $\mathcal{H}$ produced by Theorem 2.6 for a specific small reduced palette $P$; any discrepancy between the two would falsify the bridge identity. Alternatively, find a reduced palette $P$ and an integer $n$ such that some palette in $\mathrm{EX}_{\mathrm{pal}}(n, \mathcal{F}(P)_M)$ is neither a blow-up of $P$ nor of $\mathrm{rev}(P)$, which would contradict Theorem 2.6 directly.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every $\lambda \in \Lambda_{\mathrm{pal}}$ there is a finite family $\mathcal{F}$ of 3-graphs with $\pi_{\therefore}(\mathcal{F}) = \lambda$. The proof first reduces the problem to palette Turán densities, then establishes a stability theorem (Theorem 2.6): for any reduced palette $P$, one can construct a finite family $\mathcal{H}$ such that $P$ paints no member of $\mathcal{H}$, and every palette that is extremal for $\mathcal{H}$ is a blow-up of $P$ or of its reverse $\mathrm{rev}(P)$. Using the known equality between palette Turán density and uniform Turán density for finite families, the paper concludes that $\pi_{\therefore}(\mathcal{H}) = \Lambda_P$. A direct corollary is that $\Lambda_{\mathrm{pal}} \subseteq \Pi_{\therefore,\mathrm{fin}}$, so the set of uniform Turán densities of finite families contains the Lagrangians of all 3-graphs and includes irrational numbers.

Load-bearing premise

The argument assumes the previously proved identity $\pi_{\mathrm{pal}}(\mathcal{F}) = \pi_{\therefore}(\mathcal{F})$ for every finite family $\mathcal{F}$; if that identity failed for some constructed family, the main theorem would not follow from the stability result.

Editorial extensions

If this is right

  • Every $\lambda \in \Lambda_{\mathrm{pal}}$ is realized by some finite family of 3-graphs, so $\Lambda_{\mathrm{pal}} \subseteq \Pi_{\therefore,\mathrm{fin}}$.
  • Since $\Lambda_{\mathrm{pal}}$ contains irrational numbers, $\Pi_{\therefore,\mathrm{fin}}$ contains irrational numbers, settling a natural question raised by the previously known rational examples.
  • Combined with the recent approximation theorem, the set $\Pi_{\therefore,\mathrm{fin}}$ is squeezed between $\Lambda_{\mathrm{pal}}$ and its closure.
  • Every non-jump in the ordinary Turán densities of 3-graphs yields a non-jump in $\Pi_{\therefore,\mathrm{fin}}$.
  • The finite families witnessing the theorem are enormous: the Ramsey-theoretic step forces bounds of tower type, so the result is an existence statement rather than a practical construction.

Reading between the lines

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

  • The proof leaves open whether the inclusion $\Lambda_{\mathrm{pal}} \subseteq \Pi_{\therefore,\mathrm{fin}}$ is strict; if it is strict, there would be a uniform Turán density of a finite family not representable by any palette Lagrangian, which would show that the palette-construction heuristic in the area is incomplete.
  • The true bottleneck is the imported identity $\pi_{\mathrm{pal}} = \pi_{\therefore}$ for finite families; the stability part is internal, so a direct proof of that identity would remove the main ingredient the paper does not prove.
  • The same architecture—palette removal lemma, Ramsey distinction, and stability—appears transferable to $k$-uniform hypergraphs for $k>3$, provided the analogous palette removal lemma and ordered Ramsey theorem hold; nothing in the reduction seems to force $k=3$ except those tools.
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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

1 major / 5 minor

Summary. The paper proves Theorem 1.1: every value in Lambda_pal, the set of Lagrangians of finite palettes, occurs as the uniform Turán density pi_therefore(F) of some finite family F of 3-graphs. The proof introduces a palette analogue of the removal lemma, a Ramsey-theoretic statement (Lemma 4.1) showing that palettes not contained in blow-ups of P or rev(P) can be separated by a 3-graph, and a stability theorem (Theorem 2.6) asserting that for every reduced palette P there is a finite family H whose extremal palettes are exactly blow-ups of P or of rev(P). The main theorem follows by combining Theorem 2.6 with the known equality between uniform Turán density, reduced-graph Turán density, and palette Turán density for finite families (Reiher; Reiher–Rödl–Schacht; Lamaison).

Significance. This is a substantial and timely result: it establishes the exact converse direction to Lamaison's approximation theorem, showing that the palette construction is not only dense in the set of uniform Turán densities but also yields every palette Lagrangian exactly. The corollary that Pi_therefore,fin contains irrational numbers resolves a natural question about finite families. The proof is technically rich, combining regularity and removal for ordered triples, the Nešetřil–Rödl partite construction, and a Pikhurko-style stability argument; the auxiliary regularity lemmas are proved in an appendix. The paper is transparent about its use of external results, and the main derivation does not assume the target theorem.

major comments (1)
  1. [§6, proof of Theorem 2.6] The step 'Corollary 6.1 implies Q is alpha-close to being contained in a blow-up of P or rev(P); therefore, by Lemma 6.2, Q is a blow-up' contains a factor-of-two gap. With alpha = 1/2 min{alpha_P, alpha_rev}, Corollary 6.1 yields a subpalette Q' of a blow-up S with |Q \setminus Q'| <= alpha n^3. Since S is H-deficient and Q is extremal, |S \setminus Q| <= |Q \setminus S|, so |Q △ S| <= 2 alpha n^3 = min{alpha_P, alpha_rev}. Lemma 6.2, however, requires alpha_P/2- or alpha_rev/2-closeness to the blow-up itself, so the conclusion does not follow as written. The argument is repaired by applying Corollary 6.1 with alpha = 1/4 min{alpha_P, alpha_rev}, giving 2 alpha = 1/2 min{alpha_P, alpha_rev}; no other part of the proof needs to change.
minor comments (5)
  1. [Lemma 3.10] The hypothesis states that C(P) = V_1 ∪ ... ∪ V_s is a partition, but the application in Lemma 3.3 may need the same set U_i for several pairs uv; the proof's copy construction works if the V_i are merely an indexed family of subsets of C(P), so please rephrase the statement.
  2. [Definition 7.3 and proof of Theorem 7.6] The notation '(d, q)-dense' and '(pi_rd(H) - epsilon, q)-dense' contains a stray q; the intended density parameter is d or pi_rd(H) - epsilon.
  3. [§8] The sentence 'this means that Pi_therefore,fin ⊊ Lambda_pal' appears to reverse the inclusion; the announced example gives an element of Pi_therefore,fin not in Lambda_pal, so it would imply Lambda_pal ⊊ Pi_therefore,fin (or at least Pi_therefore,fin ⊄ Lambda_pal).
  4. [Proof of Theorem 2.6] The term 'R-free' is used without a definition; it should be made explicit that a palette Q is R-free if it contains no subpalette isomorphic to any R in the family R.
  5. [Proof of Lemma 6.2] Condition (a) in the choice of M says 'M >> ...'; the subsequent inequality gamma^t (beta/2)^{M1 - 2t} > alpha needed for (6.9) should be included as an explicit requirement on M1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main result is derived from an external palette/regularity bridge plus internally proved stability and Ramsey lemmas, and the only self-citations are contextual or corollary-level.

full rationale

The main theorem (Theorem 1.1) is not equivalent to its inputs by construction. The proof reduces to Theorem 2.6, which constructs a finite family H=F(P)_M from the palette P and proves via Lemma 3.3 (removal), Lemma 4.1 (Ramsey distinction), and Lemma 6.2 (stability) that every extremal palette of H is a blow-up of P or of rev(P); these lemmas are proved in the paper from external results (Nešetřil–Rödl, Kohayakawa–Nagle–Rödl–Schacht) and do not assume the target equality. The bridge π_pal(F)=π_{∴}(F) for finite F is cited from Reiher/Rödl/Schacht and Lamaison, an external theorem whose assumptions do not include the conclusion, so it is independent support rather than a circular import. The self-citation [21] is used only for contextual statements (Λ_pal⊆Π_{∴,∞} and the irrationality corollary), and Theorem 1.1 is proved without relying on it. No parameter is fitted to data and then renamed a prediction; the construction of H is direct. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new postulated entities. It defines mathematical objects (palettes, reverse palettes, rigid palettes) within the proof, but these are not invented entities in the sense of new forces or particles. The load-bearing external assumptions are listed above.

assumptions (4)
  • domain assumption Equality of palette Turan density and uniform Turan density for finite families (Theorem 7.4, cited from Reiher [31] and implicit in Reiher-Rodl-Schacht [34], with the palette form from Lamaison [24]).
    This is the bridge that converts the palette stability result into the uniform Turan density claim; the paper does not prove it.
  • standard math Nesetril-Rodl edge-Ramsey theorem for ordered linear k-graphs (Theorem 4.2, cited from [28]).
    Basis for the structural Ramsey lemma (Lemma 4.1) that distinguishes palettes.
  • standard math Szemeredi regularity lemma and counting lemma for linear hypergraphs (Lemma 3.9, cited from Kohayakawa-Nagle-Rodl-Schacht [22]).
    Used in the palette removal lemma to count paintings.
  • domain assumption Pikhurko's stability methodology [30] applies to palettes.
    The stability argument in Section 6 is adapted from Pikhurko's work on Turan densities; the paper assumes the framework transfers to the palette setting.

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Pith. "Pith review of On possible uniform Tur\'an densities." pith.science (2026). https://pith.science/paper/6LQX2IOR

@misc{pith2026250421220,
  author       = {Pith},
  title        = {Pith review of: On possible uniform Tur\'an densities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LQX2IOR}},
  note         = {Machine review of arXiv:2504.21220}
}
abstract

Given a family of $3$-graphs $\mathcal{F}$, the uniform Tur\'{a}n density $\pi_{\therefore}(\mathcal{F})$ is defined as the infimum $d\in[0,1]$ for which any sufficiently large uniformly $d$-dense $3$-graph - that is, a $3$-graph which has edge-density at least $d$ on all linearly sized subsets - contains a copy of some $F \in \mathcal{F}$. Let $\Pi_{\therefore,\text{fin}}$ denote the set of all possible uniform Tur\'{a}n densities of finite families. Erd\H{o}s, Hajnal, and R\"{o}dl introduced a family of constructions for lower bounds on uniform Tur\'an densities called palette constructions. We show that $\Pi_{\therefore,\text{fin}}$ contains every $d$ that is obtained as the uniform density of an optimized palette construction. A corollary of this is that $\Pi_{\therefore,\text{fin}}$ contains the set of Lagrangians of $3$-graphs and includes irrational numbers. Our work complements a recent result of Lamaison, which states that every value in $\Pi_{\therefore,\text{fin}}$ can be approximated by uniform densities of palette constructions.

Figures

Figures reproduced from arXiv: 2504.21220 by the authors.

Figure 4.1
Figure 4.1. An example with n “ 2. The ordered graph A0 1 is in red and A0 2 is in blue. On the second line we have the Ramsey graph A1 1 with several copies of A0 1 all intersecting in either an edge or a single vertex. Finally, in the last line we have the graph A1 2 on the same set of vertices. Claim 4.4. Let 0 ď j ď n. Then every t-coloring of An contains a copy of Aj such that the ordered graphs A j k are monochromatic for… view at source ↗
Figure 4.2
Figure 4.2. An example of Gσ for the permutation σ P S4 given by σp1q “ 3, σp2q “ 1, σp3q “ 4 and σp4q “ 2 and aσ “ 2, bσ “ 3, cσ “ 1 and dσ “ 2. The edges are given by e1 “ t1, 2, 4, 6u (green), e2 “ t3, 4, 5, 7u (blue) and e3 “ t2, 5, 8, 9u (red) [PITH_FULL_IMAGE:figures/full_fig_p016_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. An example of G for the palette Q “ tq1, q2, q3, q4u given by q1 “ pblue, green, blueq, q2 “ pblue,red,redq, q3 “ pgreen, green, blueq and q4 “ pred, blue, greenq. The graph G consists of m “ 4 triangles and it can be partitioned into Gblue Y Ggreen Y Gred as shown in the picture. We construct our desired 3-graph F p3q by applying Propositions 4.3 and 4.5. Let pH, ăq with H “ Ťn i“1 Hi be the ordered graph obtained … view at source ↗

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Forward citations

Cited by 3 Pith papers

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    Proves exact degree-square Turán formulas for tournament palettes via auxiliary digraphs and majorization, yielding finite 3-graphs with uniform densities approaching 1/3.

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Works this paper leans on

38 extracted references · 36 canonical work pages · cited by 3 Pith papers

  1. [21]

    D. King, M. Sales, and B. Schülke,Lagrangians are attained as uniform Turán densities(2024). Ò 1 , 1 , 8

    work page 2024

  2. [1]

    F. G. Abramson and L. A. Harrington,Models without indiscernibles, J. Symbolic Logic43 (1978), no. 3, 572–600. MR 503795 Ò 4

    work page 1978

  3. [2]

    N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy,Efficient testing of large graphs, Combinatorica 20 (2000), no. 4, 451–476.Ò 2 , 3 , A , A

    work page 2000

  4. [3]

    Alon and A

    N. Alon and A. Shapira,Every monotone graph property is testable, Proceedings of the thirty-seventh annual acm symposium on theory of computing, 2005, pp. 128–137.Ò 3

    work page 2005

  5. [4]

    Alon and J

    N. Alon and J. H Spencer,The probabilistic method, 4th ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Nashville, TN, 2016 (en).Ò A

    work page 2016

  6. [5]

    Avart, V

    C. Avart, V. Rödl, and M. Schacht,Every monotone 3-graph property is testable, SIAM Journal on Discrete Mathematics 21 (2007), no. 1, 73–92.Ò 3

    work page 2007

  7. [6]

    V. Bhat, J. Nešetřil, C. Reiher, and V. Rödl,A Ramsey class for Steiner systems, J. Combin. Theory Ser. A 154 (2018), 323–349. MR 3718069 Ò 4

    work page 2018

  8. [7]

    W. G. Brown and M. Simonovits,Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures, Discret. Math.48 (1984), 147–162. Ò 1

    work page 1984

Show all 38 references
  1. [8]

    Erdős and A

    P. Erdős and A. Hajnal,On Ramsey like theorems. Problems and results, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), 1972, pp. 123–140. MR 337636 Ò 2

    1972

  2. [9]

    Erdős and M

    P. Erdős and M. Simonovits,A limit theorem in graph theory, Studia Sci. Math. Hungar.1 (1966), 51–57. MR 205876 Ò 1

    1966

  3. [10]

    Erdős and V

    P. Erdős and V. T. Sós,On Ramsey-Turán type theorems for hypergraphs, Combinatorica 2 (1982), no. 3, 289–295. MR 698654 Ò 1

    1982

  4. [11]

    Erdős and A

    P. Erdős and A. H. Stone,On the structure of linear graphs, Bull. Amer. Math. Soc.52 (1946), 1087–1091. MR 18807 Ò 1

    1946

  5. [12]

    Erdős,Problems and results on graphs and hypergraphs: similarities and differences(1990), 12–28

    P. Erdős,Problems and results on graphs and hypergraphs: similarities and differences(1990), 12–28. MR 1083590 Ò 1

    1990

  6. [13]

    Erdös and G

    P. Erdös and G. Szekeres,A combinatorial problem in geometry, Compositio Math.2 (1935), 463–470. MR 1556929 Ò 4

    1935

  7. [14]

    Erdös,On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics2 (1964), no

    P. Erdös,On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics2 (1964), no. 3, 183–190.Ò 8

    1964

  8. [15]

    Frankl and V

    P. Frankl and V. Rödl,Hypergraphs do not jump, Combinatorica 4 (1984), 149–159. Ò 1 , 8

    1984

  9. [16]

    Garbe, D

    F. Garbe, D. Iľkovič, D. Kráľ, F. Kučerák, and A. Lamaison,Hypergraphs with uniform Turán density equal to 8/27(2024), available at arXiv:2407.05829 . Ò 1

    2024

  10. [17]

    Garbe, D

    F. Garbe, D. Král’, and A. Lamaison,Hypergraphs with minimum positive uniform Turán density, Israel J. Math.259 (2024), no. 2, 701–726. MR 4732978 Ò 1

    2024

  11. [18]

    Glebov, D

    R. Glebov, D. Král’, and J. Volec,A problem of Erdős and Sós on 3-graphs, Israel J. Math.211 (2016), no. 1, 349–366. MR 3474967 Ò 1 , 1

    2016

  12. [19]

    Katona, T

    G. Katona, T. Nemetz, and M. Simonovits,On a problem of Turán in the theory of graphs, Mat. Lapok 15 (1964), 228–238 (Hungarian, with English and Russian summaries). MR 172263 Ò 1 38 D. KING, S. PIGA, M. SALES, AND B. SCHÜLKE

    1964

  13. [20]

    Keevash,Hypergraph Turán problems(2011), 83–139

    P. Keevash,Hypergraph Turán problems(2011), 83–139. MR 2866732 Ò 1

    2011

  14. [22]

    Kohayakawa, B

    Y. Kohayakawa, B. Nagle, V. Rödl, and M. Schacht,Weak hypergraph regularity and linear hypergraphs, Journal of Combinatorial Theory, Series B100 (2010), no. 2, 151–160.Ò 3.9

    2010

  15. [23]

    Král’, F

    D. Král’, F. Kučerák, A. Lamaison, and G. Tardos,Uniform Turán density—palette classification. In preparation. Ò 1

  16. [24]

    Lamaison,Palettes determine uniform Turán density(2024)

    A. Lamaison,Palettes determine uniform Turán density(2024). Ò 1 , 2 , 2 , 4 , 7 , 7 , 8 , 8

    2024

  17. [25]

    Lamaison and Z

    A. Lamaison and Z. Wu,Relating the Turán density and the uniform Turán density of hypergraphs. In preparation. Ò 8

  18. [26]

    , The uniform Turán density of large stars(2024), available at arXiv:2409.03699 . Ò 1

    2024 arXiv

  19. [27]

    T. S. Motzkin and E. G. Straus,Maxima for graphs and a new proof of a theorem of Turán, Canadian J. Math. 17 (1965), 533–540. MR 0175813 Ò 1

    1965

  20. [28]

    Nešetřil and V

    J. Nešetřil and V. Rödl,Strong Ramsey theorems for Steiner systems, Trans. Amer. Math. Soc.303 (1987), no. 1, 183–192. MR 896015 Ò 1 , 4 , 4.2

    1987

  21. [29]

    , The partite construction and Ramsey set systems, 1989, pp. 327–334. Graph theory and combinatorics (Cambridge, 1988). MR 1001405 Ò 4

    1989

  22. [30]

    Pikhurko,On possible Turán densities, Israel Journal of Mathematics201 (201204)

    O. Pikhurko,On possible Turán densities, Israel Journal of Mathematics201 (201204). Ò 1 , 2 , 5 , 6 , 6 , 8

  23. [31]

    Reiher,Extremal problems in uniformly dense hypergraphs, European J

    Chr. Reiher,Extremal problems in uniformly dense hypergraphs, European J. Combin.88 (2020), 103117, 22. MR 4111729 Ò 1 , 2 , 2 , 2 , 7 , 7.4

    2020

  24. [32]

    Reiher, V

    C. Reiher, V. Rödl, and M. Schacht,Hypergraphs with vanishing Turán density in uniformly dense hypergraphs, J. Lond. Math. Soc. (2)97 (2018), no. 1, 77–97. MR 3764068 Ò 1

    2018

  25. [33]

    Reiher and V

    C. Reiher and V. Rödl,The girth ramsey theorem(2023). Ò 4

    2023

  26. [34]

    Reiher, V

    Chr. Reiher, V. Rödl, and M. Schacht,On a Turán problem in weakly quasirandom 3-uniform hypergraphs, J. Eur. Math. Soc. (JEMS)20 (2018), no. 5, 1139–1159. MR 3790065 Ò 1 , 1 , 7 , 7.4 , 8

    2018

  27. [35]

    Rödl,On universality of graphs with uniformly distributed edges, Discrete Math.59 (1986), no

    V. Rödl,On universality of graphs with uniformly distributed edges, Discrete Math.59 (1986), no. 1-2, 125–134. MR 837962 Ò 2

    1986

  28. [36]

    Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat

    P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452 (Hungarian, with German summary). MR 18405 Ò 1 Appendix A. Weak Palette Regularity Before we begin with details we provide a brief overview. Our proof of Theorem 3.7 will be as expected. ...

    1941

  29. [37]

    □ Finally, the following lemma will be used to redistribute the exceptional setV0 amongst the other classes of the equipartition without destroying regularity

    Overcounting the number of vertices discarded gives|U0|ď| V0|` n 2t, and flattening pairspj,kq to a single indexℓ gives an equipartition B with at most2t23t2 2tmembers, as needed. □ Finally, the following lemma will be used to redistribute the exceptional setV0 amongst the oth...

  30. [38]

    Then with probability at least1 2 both Part (iii) and (iv) are satisfied as well, so there exists such a choice ofUi and we are done. □ Mathematics Department, California Institute of Technology, Pasadena, USA Email address: dking@caltech.edu F achbereich Mathematik, Universit...

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