An intersection theorem for systems of sets

Kostochka, Alexandr

doi:10.1002/(sici)1098-2418(199608/09)9:1/2<213::aid-rsa13>3.0.co;2-o

Cited by 11 publications

(6 citation statements)

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“…With regard to the earlier results in [7], Noga Alon kindly pointed out to us that Lemma 5 of [7] is actually a version of a result of Erdős and Rado [9] on ∆-systems (or sunflowers). Small quantitative improvements here are due to Kostochka [12], which together with the previous argument, for the set S 2 of squares, would lead to the tiny improvement d = O((log 2 N ) 2 log 5 N log 4 N ), the log i N denoting the i-fold iterated logarithm. Moreover, the Erdős-Rado conjecture on these ∆-systems, for which Erdős [8] offered a prize of $1000, would have implied d = O(log log N ).…”

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confidence: 59%

Hilbert cubes in progression-free sets and in the set of squares

Dietmann

Elsholtz

2012

Isr. J. Math.

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Let S 2 be the set of integer squares. We show that the dimension d of a Hilbert cube a 0 + {0, a 1 } + · · · + {0, a d } ⊂ S 2 is bounded by d = O(log log N ). Letdenote a Hilbert cube of dimension d. Brown, Erdős and Freedman [2] asked whether the maximal dimension of a Hilbert cube in the set of squares is absolutely bounded or not. Experimentally, one finds only very small cubes such asObserve that in this example 29 2 and 37 2 occur as sums in two different ways. Cilleruelo and Granville [3] and Solymosi [15] and Alon, Angel, Benjamini and Lubetzky [1] explain that the Bombieri-Lang conjecture implies that d is absolutely bounded. Hegyvári and Sárközy ([11], Theorem 1) proved that for the set of integer squares S 2 ∩ [1, N ] the maximal dimension is bounded by d = O((log N ) 1/3 ). Dietmann and Elsholtz ([7], Theorem 3) improved this to d = O((log log N ) 2 ). Here we further reduce that bound.Theorem 1 (Main theorem). Let S 2 denote the set of integer squares. Let N be sufficiently large, let a 0 be a non-negative integer and let A = {a 1 , . . . , a d } be a set of distinct positive integers such that H(a 0 ; a 1 , . . . , a d ) ⊆ S 2 ∩ [1, N ]. Then d ≤ 7 log log N.A comparable bound was proved in the author's earlier paper ([7], Theorem 1) for higher powers instead of squares.Remark. The special case of subsetsums, i.e. Hilbert cubes with a 0 = 0, was previously studied by Csikvári ([5], Corollary 2.5), who proved in this case the same bound d = O(log log N ). His method of proof would not extend to the general case of a 0 = 0.Corollary 1. Let f (x) = ax 2 + bx + c be a quadratic polynomial where a, b, c ∈ Z such that a > 0, and let S = {f (x) : x ∈ N}. Let a 0 be a non-negative integer and

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confidence: 59%

Hilbert cubes in progression-free sets and in the set of squares

Dietmann

Elsholtz

2012

Isr. J. Math.

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“…Theorem 2.6 (Kostochka [11]). For p ≥ 3 and α > 1, there exists D(p, α) such that q(k, p) ≤ D(p, α)k!…”

Section: Preliminariesmentioning

confidence: 99%

Regular subgraphs of uniform hypergraphs

Kim

2016

Journal of Combinatorial Theory, Series B

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We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if $r\in\{3,4\}$, $r\mid k$ and $k,n$ are both sufficiently large, then the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $r$-regular subgraphs is exactly ${{n-1} \choose {k-1}}$, with equality only if all edges contain a specific vertex $v$. We also ask some related questions

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“…With regard to our earlier result, Noga Alon kindly pointed out to us that Lemma 5 of [11], which corresponds to Lemma 9 in this paper, is actually a version of a result of Erdős and Rado [14] on ∆-systems (or sunflowers). On this subject small quantitative improvements are due to Kostochka [35], even though the explicit dependence on the parameters h and v is not well understood, and apparently at least one of the parameters h and v is fixed. For the set S 2 of squares, this would, assuming uniformity, possibly lead to the tiny improvement d = O((log 2 N ) 2 log 5 N log 4 N ), the log i N denoting the i-fold iterated logarithm.…”

Section: 1mentioning

confidence: 99%

Hilbert cubes in arithmetic sets

Dietmann

Elsholtz

2015

Rev. Mat. Iberoam.

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We show upper bounds on the maximal dimension d of Hilbert cubes H = a 0 +{0, a 1 }+· · ·+{0, a d } ⊂ S ∩[1, N ] in several sets S of arithmetic interest. a) For the set of squares we obtain d = O(log log N ). Using previously known methods this bound could have been achieved only conditionally subject to an unsolved problem of Erdős and Rado. b) For the set W of powerful numbers we show d = O((log N ) 2 ). c) For the set V of pure powers we also show d = O((log N ) 2 ), but for a homogeneous Hilbert cube, with a 0 = 0, this can be improved to d = O((log log N ) 3 / log log log N ), when the a i are distinct, and d = O((log log N ) 4 /(log log log N ) 2 ), generally. This compares with a result of d = O((log N ) 3 /(log log N ) 1/2 ) in the literature. d) For the set V we also solve an open problem of Hegyvári and Sárközy, namely we show that V does not contain an infinite Hilbert cube. e) For a set without arithmetic progressions of length k we prove d = O k (log N ), which is close to the true order of magnitude.

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An intersection theorem for systems of sets

Cited by 11 publications

References 6 publications

Hilbert cubes in progression-free sets and in the set of squares

Hilbert cubes in progression-free sets and in the set of squares

Regular subgraphs of uniform hypergraphs

Hilbert cubes in arithmetic sets

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