The Erdos-Rado sunflower lemma - Revision history

Teorth: /* Small values */

2020-07-29T21:26:01Z

Small values

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	Below is a collection of known constructions for small values, taken from Abbott-Exoo.		Below is a collection of known constructions for small values, taken from Abbott-Exoo.
	Boldface stands for matching upper bound (and best known upper bounds are planned to be added to other entries).		Boldface stands for matching upper bound (and best known upper bounds are planned to be added to other entries).
	Also note that for <math>k</math> fixed we have <math>f(k,r)=k^r+o(k^r)</math> from Kostochka-Rödl-Talysheva.		Also note that for <math>k</math> fixed we have <math>f(k,r)=r^k+o(r^k)</math> from Kostochka-Rödl-Talysheva (note that that paper swaps the notation for k and r).

	{\| class="wikitable"		{\| class="wikitable"

Domotorp: /* Variants and notation */

2016-02-05T18:58:55Z

Variants and notation

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	<B>Disproven for <math>k=3,r=3</math></B>: set <math>\|V_1\|=\|V_2\|=\|V_3\|=3</math> and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.		<B>Disproven for <math>k=3,r=3</math></B>: set <math>\|V_1\|=\|V_2\|=\|V_3\|=3</math> and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.

	A ''weak sunflower'' (''weak Delta-system'') of size <math>r</math> is a family of <math>r</math> sets, <math> A_1,\ldots,A_r</math>, such that their pairwise intersections have the same size, i.e., <math> \|A_i\cap A_j\|=\|A_{i'}\cap A_{j'}\|</math> for every <math> i\ne j</math> and <math> i'\ne j'</math>. If we denote the size of the largest family of <math>k</math>-sets without an <math>r</math>-weak sunflower by <math>g(k,r)</math>, by definition we have <math>g(k,r)\le f(k,r)</math>. ~~Also, if~~ we ~~denote by~~ <math>~~R_r~~(k)~~-1</math> the size of the largest complete graph whose edges can be colored with <math>r</math> colors such that there is no monochromatic clique on <math>k</math> vertices, then we have <math>~~g(k,r)~~\le R_r~~(k)-1</math>, as ~~we can color the edges running between the <math>k</math>~~-~~sets of our weak sunflower~~-~~free family with the intersection sizes. For all three functions only exponential lower bounds and factorial type upper bounds are known~~.		A ''weak sunflower'' (''weak Delta-system'') of size <math>r</math> is a family of <math>r</math> sets, <math> A_1,\ldots,A_r</math>, such that their pairwise intersections have the same size, i.e., <math> \|A_i\cap A_j\|=\|A_{i'}\cap A_{j'}\|</math> for every <math> i\ne j</math> and <math> i'\ne j'</math>. If we denote the size of the largest family of <math>k</math>-sets without an <math>r</math>-weak sunflower by <math>g(k,r)</math>, by definition we have <math>g(k,r)\le f(k,r)</math>. For this function we also know that <math> g(a+b,r)\ge g(a,r)g(b,r)</math>, as shown by Abbott-Hanson in On finite Δ-systems II.

	Denote by <math>3DES(n)</math> the largest integer such that there is a group of size <math>n</math> and a subset <math>S</math> of size <math>3DES(n)</math> without three ''disjoint equivoluminous subsets'', i.e., there is no <math>S=S_1\cup^* S_2\cup^* S_3\cup^* S_{rest}</math> such that <math>\sum_{s\in S_1} s=\sum_{s\in S_2} s=\sum_{s\in S_3} s</math>. ~~Then~~ <math>{3DES(n) \choose DES(n)} / n \le g(DES(n),3)</math> holds, thus if <math>g(k,3)</math> grows exponentially, then <math>3DES(n)=O(\log n)</math>.		Also, if we denote by <math>R_r(k)-1</math> the size of the largest complete graph whose edges can be colored with <math>r</math> colors such that there is no monochromatic clique on <math>k</math> vertices, then we have <math>g(k,r)\le R_r(k)-1</math>, as we can color the edges running between the <math>k</math>-sets of our weak sunflower-free family with the intersection sizes. For all three functions only exponential lower bounds and factorial type upper bounds are known.

			Denote by <math>3DES(n)</math> the largest integer such that there is a group of size <math>n</math> and a subset <math>S</math> of size <math>3DES(n)</math> without three ''disjoint equivoluminous subsets'', i.e., there is no <math>S=S_1\cup^* S_2\cup^* S_3\cup^* S_{rest}</math> such that <math>\sum_{s\in S_1} s=\sum_{s\in S_2} s=\sum_{s\in S_3} s</math>. As noticed by Alon-Shpilka-Umans, <math>{3DES(n) \choose DES(n)} / n \le g(DES(n),3)</math> holds, thus if <math>g(k,3)</math> grows exponentially, then <math>3DES(n)=O(\log n)</math>.

	== Small values ==		== Small values ==

Domotorp: /* Bibliography */ added alon-shpilka-umans

2016-02-05T18:54:41Z

Bibliography: added alon-shpilka-umans

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	* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz		* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz
	* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).		* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).
			* [http://users.cms.caltech.edu/~umans/papers/ASU11-final.pdf On Sunflowers and Matrix Multiplication], N. Alon, A. Shpilka, C. Umans, Computational Complexity 22 (2013), 219-243.
	* [http://www.sciencedirect.com/science/article/pii/0012365X9400185L On set systems without weak 3-Δ-subsystems], M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.		* [http://www.sciencedirect.com/science/article/pii/0012365X9400185L On set systems without weak 3-Δ-subsystems], M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.
	* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12 (1961), 313–320.		* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12 (1961), 313–320.

Domotorp: /* Variants and notation */ cleared up DES part

2016-02-05T16:15:39Z

Variants and notation: cleared up DES part

← Older revision		Revision as of 09:15, 5 February 2016
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	<B>Disproven for <math>k=3,r=3</math></B>: set <math>\|V_1\|=\|V_2\|=\|V_3\|=3</math> and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.		<B>Disproven for <math>k=3,r=3</math></B>: set <math>\|V_1\|=\|V_2\|=\|V_3\|=3</math> and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.

	A ''weak sunflower'' (''weak Delta-system'') of size <math>r</math> is a family of <math>r</math> sets, <math> A_1,\ldots,A_r</math>, such that their pairwise intersections have the same size, i.e., <math> \|A_i\cap A_j\|=\|A_{i'}\cap A_{j'}\|</math> for every <math> i\ne j</math> and <math> i'\ne j'</math>. If we denote the size of the largest family of <math>k</math>-sets without an <math>r</math>-weak sunflower by <math>g(k,r)</math>, by definition we have <math>g(k,r)\le f(k,r)</math>. Also, if we denote by <math>R_r(k)-1</math> the size of the largest complete graph whose edges can be colored with <math>r</math> colors such that there is no monochromatic clique on <math>k</math> vertices, then we have <math>g(k,r)\le R_r(k)-1</math>, as we can color the edges running between the <math>k</math>-sets of our weak sunflower-free family with the intersection sizes. ~~Also, denote~~ by <math>3DES(n)</math> the ~~least~~ integer such that ~~given~~ <math>~~3DES(~~n)</math> ~~elements~~ <math>S</math> ~~from a group~~ of size <math>n</math>~~, one can always select~~ three disjoint equivoluminous ~~subset of <math>S</math>~~, i.e., there ~~exists~~ <math>S=S_1\cup^* S_2\cup^* S_3\cup^* ~~S_0~~</math> such that <math>\sum_{s\in S_1}=\sum_{s\in S_2}=\sum_{s\in S_3}</math>. Then if <math>2^{3DES(n)}/n></math> ~~- to be continued..~~.		A ''weak sunflower'' (''weak Delta-system'') of size <math>r</math> is a family of <math>r</math> sets, <math> A_1,\ldots,A_r</math>, such that their pairwise intersections have the same size, i.e., <math> \|A_i\cap A_j\|=\|A_{i'}\cap A_{j'}\|</math> for every <math> i\ne j</math> and <math> i'\ne j'</math>. If we denote the size of the largest family of <math>k</math>-sets without an <math>r</math>-weak sunflower by <math>g(k,r)</math>, by definition we have <math>g(k,r)\le f(k,r)</math>. Also, if we denote by <math>R_r(k)-1</math> the size of the largest complete graph whose edges can be colored with <math>r</math> colors such that there is no monochromatic clique on <math>k</math> vertices, then we have <math>g(k,r)\le R_r(k)-1</math>, as we can color the edges running between the <math>k</math>-sets of our weak sunflower-free family with the intersection sizes. For all three functions only exponential lower bounds and factorial type upper bounds are known.

			Denote by <math>3DES(n)</math> the largest integer such that there is a group of size <math>n</math> and a subset <math>S</math> of size <math>3DES(n)</math> without three ''disjoint equivoluminous subsets'', i.e., there is no <math>S=S_1\cup^* S_2\cup^* S_3\cup^* S_{rest}</math> such that <math>\sum_{s\in S_1} s=\sum_{s\in S_2} s=\sum_{s\in S_3} s</math>. Then <math>{3DES(n) \choose DES(n)} / n \le g(DES(n),3)</math> holds, thus if <math>g(k,3)</math> grows exponentially, then <math>3DES(n)=O(\log n)</math>.

	== Small values ==		== Small values ==

Domotorp: /* Variants and notation */ added weak sunflower section

2016-02-05T15:56:10Z

Variants and notation: added weak sunflower section

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	== Variants and notation ==		== Variants and notation ==

	Given a family F of sets and a set S, the <I>star</I> of S is the subfamily of those sets in F containing S, and the <I>link</I> of S is obtained from the star of S by deleting the elements of S from every set in the star. (We use the terms link and star because we do want to consider eventually hypergraphs as geometric/topological objects.)		Given a family <math>\cal F</math> of sets and a set S, the <I>star</I> of S is the subfamily of those sets in <math>\cal F</math> containing S, and the <I>link</I> of S is obtained from the star of S by deleting the elements of S from every set in the star. (We use the terms link and star because we do want to consider eventually hypergraphs as geometric/topological objects.)

	We can restate the delta system problem as follows: f(k,r) is the maximum size of a family of k-sets such that the link of every set A does not contain r pairwise disjoint sets.		We can restate the delta system problem as follows: f(k,r) is the maximum size of a family of k-sets such that the link of every set A does not contain r pairwise disjoint sets.
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	<B>Reduction (folklore)</B>: It is enough to prove Erdos-Rado Delta-system conjecture for the balanced case.		<B>Reduction (folklore)</B>: It is enough to prove Erdos-Rado Delta-system conjecture for the balanced case.

	<B>Proof</B>: Divide the elements into d color classes at random and take only colorful sets. The expected size of the surviving colorful sets is k!/k^k\|F\|.		<B>Proof</B>: Divide the elements into d color classes at random and take only colorful sets. The expected size of the surviving colorful sets is <math>k!/k^k \cdot \|\cal F\|</math>.

	<B>Hyperoptimistic conjecture:</B> The maximum size of a balanced collection of k-sets without a sunflower of size r is (r-1)^k.		<B>Hyperoptimistic conjecture:</B> The maximum size of a balanced collection of k-sets without a sunflower of size r is (r-1)^k.

	<B>Disproven for <math>k=3,r=3</math></B>: set <math>\|V_1\|=\|V_2\|=\|V_3\|=3</math> and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.		<B>Disproven for <math>k=3,r=3</math></B>: set <math>\|V_1\|=\|V_2\|=\|V_3\|=3</math> and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.

			A ''weak sunflower'' (''weak Delta-system'') of size <math>r</math> is a family of <math>r</math> sets, <math> A_1,\ldots,A_r</math>, such that their pairwise intersections have the same size, i.e., <math> \|A_i\cap A_j\|=\|A_{i'}\cap A_{j'}\|</math> for every <math> i\ne j</math> and <math> i'\ne j'</math>. If we denote the size of the largest family of <math>k</math>-sets without an <math>r</math>-weak sunflower by <math>g(k,r)</math>, by definition we have <math>g(k,r)\le f(k,r)</math>. Also, if we denote by <math>R_r(k)-1</math> the size of the largest complete graph whose edges can be colored with <math>r</math> colors such that there is no monochromatic clique on <math>k</math> vertices, then we have <math>g(k,r)\le R_r(k)-1</math>, as we can color the edges running between the <math>k</math>-sets of our weak sunflower-free family with the intersection sizes. Also, denote by <math>3DES(n)</math> the least integer such that given <math>3DES(n)</math> elements <math>S</math> from a group of size <math>n</math>, one can always select three disjoint equivoluminous subset of <math>S</math>, i.e., there exists <math>S=S_1\cup^* S_2\cup^* S_3\cup^* S_0</math> such that <math>\sum_{s\in S_1}=\sum_{s\in S_2}=\sum_{s\in S_3}</math>. Then if <math>2^{3DES(n)}/n></math> - to be continued...

	== Small values ==		== Small values ==

Domotorp: /* Bibliography */ fixed deltas

2016-02-05T12:56:12Z

Bibliography: fixed deltas

Domotorp: /* Bibliography */ added links

2016-02-05T12:53:25Z

Bibliography: added links

Domotorp: /* Bibliography */ added Abbott-Exoo II and Axenovich et al

2016-02-05T12:44:30Z

Bibliography: added Abbott-Exoo II and Axenovich et al

← Older revision		Revision as of 05:44, 5 February 2016
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	* On set systems not containing delta systems, H. L. Abbott and G. Exoo, Graphs and Combinatorics 8 (1992), 1–9.		* On set systems not containing delta systems, H. L. Abbott and G. Exoo, Graphs and Combinatorics 8 (1992), 1–9.
	* On finite <math>\Delta</math>-systems, H. L. Abbott and D. Hanson, Discrete Math. 8 (1974), 1-12.		* On finite <math>\Delta</math>-systems, H. L. Abbott and D. Hanson, Discrete Math. 8 (1974), 1-12.
			* On finite <math>\Delta</math>-systems II, H. L. Abbott and D. Hanson, Discrete Math. 17 (1977), 121-126.
	* Intersection theorems for systems of sets, H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389.		* Intersection theorems for systems of sets, H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389.
	* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz		* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz
	* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).		* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).
			* On set systems without weak 3-<math>\Delta</math>-subsystems, M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.
	* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12: 313–320 (1961), doi:10.1093/qmath/12.1.313.		* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12: 313–320 (1961), doi:10.1093/qmath/12.1.313.
	* [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.		* [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.

Teorth: /* Threads */

2016-01-31T16:30:16Z

Threads

← Older revision		Revision as of 09:30, 31 January 2016
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	* [https://gilkalai.wordpress.com/2015/11/03/polymath10-the-erdos-rado-delta-system-conjecture Polymath10: The Erdos Rado Delta System Conjecture], Gil Kalai, Nov 2, 2015. <I>Inactive</I>		* [https://gilkalai.wordpress.com/2015/11/03/polymath10-the-erdos-rado-delta-system-conjecture Polymath10: The Erdos Rado Delta System Conjecture], Gil Kalai, Nov 2, 2015. <I>Inactive</I>
	* [https://gilkalai.wordpress.com/2015/11/11/polymath10-post-2-homological-approach/ Polymath10, Post 2: Homological Approach], Gil Kalai, Nov 10, 2015. <I>Inactive</I>		* [https://gilkalai.wordpress.com/2015/11/11/polymath10-post-2-homological-approach/ Polymath10, Post 2: Homological Approach], Gil Kalai, Nov 10, 2015. <I>Inactive</I>
	* [https://gilkalai.wordpress.com/2015/12/08/polymath-10-post-3-how-are-we-doing/ Polymath 10 Post 3: How are we doing?], Gil Kalai, Dec 8, 2015. <B>Active</B>		* [https://gilkalai.wordpress.com/2015/12/08/polymath-10-post-3-how-are-we-doing/ Polymath 10 Post 3: How are we doing?], Gil Kalai, Dec 8, 2015. <I>Inactive</I>
			* [https://gilkalai.wordpress.com/2016/01/31/polymath10-post-4-back-to-the-drawing-board/ Polymath10-post 4: Back to the drawing board?], Gil Kalai, Jan 31, 2016. <B>Active</B>

	== External links ==		== External links ==

Teorth: /* Threads */

2015-12-08T16:51:45Z

Threads

← Older revision		Revision as of 09:51, 8 December 2015
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	* [https://gilkalai.wordpress.com/2015/11/03/polymath10-the-erdos-rado-delta-system-conjecture Polymath10: The Erdos Rado Delta System Conjecture], Gil Kalai, Nov 2, 2015. <I>Inactive</I>		* [https://gilkalai.wordpress.com/2015/11/03/polymath10-the-erdos-rado-delta-system-conjecture Polymath10: The Erdos Rado Delta System Conjecture], Gil Kalai, Nov 2, 2015. <I>Inactive</I>
	* [https://gilkalai.wordpress.com/2015/11/11/polymath10-post-2-homological-approach/ Polymath10, Post 2: Homological Approach], Gil Kalai, Nov 10, 2015. <B>Active</B>		* [https://gilkalai.wordpress.com/2015/11/11/polymath10-post-2-homological-approach/ Polymath10, Post 2: Homological Approach], Gil Kalai, Nov 10, 2015. <I>Inactive</I>
			* [https://gilkalai.wordpress.com/2015/12/08/polymath-10-post-3-how-are-we-doing/ Polymath 10 Post 3: How are we doing?], Gil Kalai, Dec 8, 2015. <B>Active</B>

	== External links ==		== External links ==

← Older revision		Revision as of 05:56, 5 February 2016
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	* [http://anothersample.net/on-set-systems-not-containing-delta-systems On set systems not containing delta systems], H. L. Abbott and G. Exoo, Graphs and Combinatorics 8 (1992), 1–9.		* [http://anothersample.net/on-set-systems-not-containing-delta-systems On set systems not containing delta systems], H. L. Abbott and G. Exoo, Graphs and Combinatorics 8 (1992), 1–9.
	* [http://www.sciencedirect.com/science/article/pii/0012365X74901034 On finite ~~<math>\Delta</math>~~-systems], H. L. Abbott and D. Hanson, Discrete Math. 8 (1974), 1-12.		* [http://www.sciencedirect.com/science/article/pii/0012365X74901034 On finite Δ-systems], H. L. Abbott and D. Hanson, Discrete Math. 8 (1974), 1-12.
	* [http://www.sciencedirect.com/science/article/pii/0012365X7790139X On finite ~~<math>\Delta</math>~~-systems II], H. L. Abbott and D. Hanson, Discrete Math. 17 (1977), 121-126.		* [http://www.sciencedirect.com/science/article/pii/0012365X7790139X On finite Δ-systems II], H. L. Abbott and D. Hanson, Discrete Math. 17 (1977), 121-126.
	* [http://www.sciencedirect.com/science/article/pii/0097316572901033 Intersection theorems for systems of sets], H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389.		* [http://www.sciencedirect.com/science/article/pii/0097316572901033 Intersection theorems for systems of sets], H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389.
	* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz		* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz
	* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).		* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).
	* [http://www.sciencedirect.com/science/article/pii/0012365X9400185L On set systems without weak 3-~~<math>\Delta</math>~~-subsystems], M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.		* [http://www.sciencedirect.com/science/article/pii/0012365X9400185L On set systems without weak 3-Δ-subsystems], M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.
	* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12~~: 313–320~~ (1961), ~~doi:10.1093/qmath/12.1.313~~.		* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12 (1961), 313–320.
	* [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.		* [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.
	* [http://arxiv.org/abs/1205.6847 On the Maximum Number of Edges in a Hypergraph with Given Matching Number], P. Frankl		* [http://arxiv.org/abs/1205.6847 On the Maximum Number of Edges in a Hypergraph with Given Matching Number], P. Frankl
	* [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.300.1783&rep=rep1&type=pdf An intersection theorem for systems of sets], A. V. Kostochka, Random Structures and Algorithms, 9 (1996), 213-221.		* [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.300.1783&rep=rep1&type=pdf An intersection theorem for systems of sets], A. V. Kostochka, Random Structures and Algorithms, 9 (1996), 213-221.
	* [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on ~~<math>\Delta</math>~~-systems], A. V. Kostochka		* [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on Δ-systems], A. V. Kostochka
	* On Systems of Small Sets with No Large Δ-Subsystems, A. V. Kostochka, V. Rödl, and L. A. Talysheva, Comb. Probab. Comput. 8 (1999), 265-268.		* On Systems of Small Sets with No Large Δ-Subsystems, A. V. Kostochka, V. Rödl, and L. A. Talysheva, Comb. Probab. Comput. 8 (1999), 265-268.
	* [http://arxiv.org/abs/1202.4196 On Erdos' extremal problem on matchings in hypergraphs], T. Luczak, K. Mieczkowska		* [http://arxiv.org/abs/1202.4196 On Erdos' extremal problem on matchings in hypergraphs], T. Luczak, K. Mieczkowska
	* Intersection theorems for systems of sets, J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.		* Intersection theorems for systems of sets, J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.

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	Edits to improve the bibliography (by adding more links, Mathscinet numbers, bibliographic info, etc.) are welcome!		Edits to improve the bibliography (by adding more links, Mathscinet numbers, bibliographic info, etc.) are welcome!

	* On set systems not containing delta systems, H. L. Abbott and G. Exoo, Graphs and Combinatorics 8 (1992), 1–9.		* [http://anothersample.net/on-set-systems-not-containing-delta-systems On set systems not containing delta systems], H. L. Abbott and G. Exoo, Graphs and Combinatorics 8 (1992), 1–9.
	* On finite <math>\Delta</math>-systems, H. L. Abbott and D. Hanson, Discrete Math. 8 (1974), 1-12.		* [http://www.sciencedirect.com/science/article/pii/0012365X74901034 On finite <math>\Delta</math>-systems], H. L. Abbott and D. Hanson, Discrete Math. 8 (1974), 1-12.
	* On finite <math>\Delta</math>-systems II, H. L. Abbott and D. Hanson, Discrete Math. 17 (1977), 121-126.		* [http://www.sciencedirect.com/science/article/pii/0012365X7790139X On finite <math>\Delta</math>-systems II], H. L. Abbott and D. Hanson, Discrete Math. 17 (1977), 121-126.
	* Intersection theorems for systems of sets, H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389.		* [http://www.sciencedirect.com/science/article/pii/0097316572901033 Intersection theorems for systems of sets], H. L. Abbott, D. Hanson, and N. Sauer, J. Comb. Th. Ser. A 12 (1972), 381–389.
	* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz		* [http://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries], Karim Adiprasito, June Huh, and Erick Katz
	* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).		* [https://www.math.uni-bielefeld.de/ahlswede/homepage/public/114.pdf The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets], R. Ahlswede, L. Khachatrian, Journal of Combinatorial Theory, Series A 76, 121-138 (1996).
	* On set systems without weak 3-<math>\Delta</math>-subsystems, M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.		* [http://www.sciencedirect.com/science/article/pii/0012365X9400185L On set systems without weak 3-<math>\Delta</math>-subsystems], M. Axenovich, D. Fon-Der-Flaassb, A. Kostochka, Discrete Math. 138 (1995), 57-62.
	* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12: 313–320 (1961), doi:10.1093/qmath/12.1.313.		* [http://www.renyi.hu/~p_erdos/1961-07.pdf Intersection theorems for systems of finite sets], P. Erdős, C. Ko, R. Rado, The Quarterly Journal of Mathematics. Oxford. Second Series 12: 313–320 (1961), doi:10.1093/qmath/12.1.313.
	* [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.		* [http://renyi.hu/~p_erdos/1960-04.pdf Intersection theorems for systems of sets], P. Erdős, R. Rado, Journal of the London Mathematical Society, Second Series 35 (1960), 85–90.
	* [http://arxiv.org/abs/1205.6847 On the Maximum Number of Edges in a Hypergraph with Given Matching Number], P. Frankl		* [http://arxiv.org/abs/1205.6847 On the Maximum Number of Edges in a Hypergraph with Given Matching Number], P. Frankl
	* An intersection theorem for systems of sets, A. V. Kostochka, Random Structures and Algorithms, 9 (1996), 213-221.		* [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.300.1783&rep=rep1&type=pdf An intersection theorem for systems of sets], A. V. Kostochka, Random Structures and Algorithms, 9 (1996), 213-221.
	* [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on <math>\Delta</math>-systems], A. V. Kostochka		* [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on <math>\Delta</math>-systems], A. V. Kostochka
	* On Systems of Small Sets with No Large Δ-Subsystems, A. V. Kostochka, V. Rödl, and L. A. Talysheva, Comb. Probab. Comput. 8 (1999), 265-268.		* On Systems of Small Sets with No Large Δ-Subsystems, A. V. Kostochka, V. Rödl, and L. A. Talysheva, Comb. Probab. Comput. 8 (1999), 265-268.
	* [http://arxiv.org/abs/1202.4196 On Erdos' extremal problem on matchings in hypergraphs], T. Luczak, K. Mieczkowska		* [http://arxiv.org/abs/1202.4196 On Erdos' extremal problem on matchings in hypergraphs], T. Luczak, K. Mieczkowska
	* Intersection theorems for systems of sets, J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.		* Intersection theorems for systems of sets, J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.