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probably many thms and applications of math that dont explicitly refer to hypergraphs are actually related to them implicitly & could be recast in those terms. because hypergraphs are equivalently just "sets of sets". in this way they're also often interchangeable with/analogous to a 2d boolean array in computer science (and how ubiquitous is that structure in both software and mathematics? in computer science it might be referred to as a "design pattern" or even just a simple "discrete structure").

here is one key appearance/application of hypergraphs not mentioned so far. the erdos-rado sunflower lemmas[1], a key discovery of extremal graph/set theory, are about an intrinsic order or emergent "structure" to "random" hypergraphs if certain somewhat modest constraints are satisfied. this lemma shows up in numerous important lower bounds proofs in monotone circuit theory in computer science, including new versions that strengthen or generalize the lemma.[2]

because of their particular role in these "bottleneck"-type proofs its not outlandish to conjecture that variations might be crucial in some future-established comp sci complexity class separations.

[1] erdos-rado sunflowers survey/refs, TCS.se

[2] The Monotone Complexity of k-Clique on Random Graphs by Rossman, containing new stronger lemmas on "quasi sunflowers"

[3] Razborov's method of approximations by WT Gowers

Probably many theorems and applications of math that don't explicitly refer to hypergraphs are actually related to them implicitly & could be recast in those terms. Because hypergraphs are equivalently just "sets of sets". In this way they're also often interchangeable with/analogous to a 2d boolean array in computer science (and how ubiquitous is that structure in both software and mathematics? In computer science it might be referred to as a "design pattern" or even just a simple "discrete structure").

Here is one key appearance/application of hypergraphs not mentioned so far. The Erdős–Rado sunflower lemmas[1], a key discovery of extremal graph/set theory, are about an intrinsic order or emergent "structure" to "random" hypergraphs if certain somewhat modest constraints are satisfied. These lemmas show up in numerous important lower bounds proofs in monotone circuit theory in computer science, including new versions that strengthen or generalize the lemma.[2]

Because of their particular role in these "bottleneck"-type proofs it's not outlandish to conjecture that variations might be crucial in some future-established computer science complexity class separations.

[1] Erdős–Rado sunflowers survey/refs, TCS.se

[2] The Monotone Complexity of $k$-Clique on Random Graphs by Rossman, containing new stronger lemmas on "quasi sunflowers"

[3] Razborov's method of approximations by WT Gowers

probably many thms and applications of math that dont explicitly refer to hypergraphs are actually related to them implicitly & could be recast in those terms. because hypergraphs are equivalently just "sets of sets". in this way they're also often interchangeable with/analogous to a 2d boolean array in computer science (and how ubiquitous is that structure in both software and mathematics? in computer science it might be referred to as a "design pattern" or even just a simple "discrete structure").

here is one key appearance/application of hypergraphs not mentioned so far. the erdos-rado sunflower lemmas[1], a key discovery of extremal graph/set theory, are about an intrinsic order or emergent "structure" to "random" hypergraphs if certain somewhat modest constraints are satisfied. this lemma shows up in numerous important lower bounds proofs in monotone circuit theory in computer science, including new versions that strengthen or generalize the lemma.[2]

because of their particular role in these "bottleneck"-type proofs its not outlandish to conjecture that variations might be crucial in some future-established comp sci complexity class separations.

[1] erdos-rado sunflowers survey/refs, TCS.se

[2] The Monotone Complexity of k-Clique on Random Graphs by Rossman, containing new stronger lemmas on "quasi sunflowers"

[3] Razborovs method of approximations by WT Gowers

probably many thms and applications of math that dont explicitly refer to hypergraphs are actually related to them implicitly & could be recast in those terms. because hypergraphs are equivalently just "sets of sets". in this way they're also often interchangeable with/analogous to a 2d boolean array in computer science (and how ubiquitous is that structure in both software and mathematics? in computer science it might be referred to as a "design pattern" or even just a simple "discrete structure").

here is one key appearance/application of hypergraphs not mentioned so far. the erdos-rado sunflower lemmas[1], a key discovery of extremal graph/set theory, are about an intrinsic order or emergent "structure" to "random" hypergraphs if certain somewhat modest constraints are satisfied. this lemma shows up in numerous important lower bounds proofs in monotone circuit theory in computer science, including new versions that strengthen or generalize the lemma.[2]

because of their particular role in these "bottleneck"-type proofs its not outlandish to conjecture that variations might be crucial in some future-established comp sci complexity class separations.

[1] erdos-rado sunflowers survey/refs, TCS.se

[2] The Monotone Complexity of k-Clique on Random Graphs by Rossman, containing new stronger lemmas on "quasi sunflowers"

[3] Razborov's method of approximations by WT Gowers

probably many thms and applications of math that dont explicitly refer to hypergraphs are actually related to them implicitly & could be recast in those terms. because hypergraphs are equivalently just "sets of sets". in this way they're also often interchangeable with/analogous to a 2d boolean array in computer science (and how ubiquitous is that structure in both software and mathematics? in computer science it might be referred to as a "design pattern" or even just a simple "discrete structure").

here is one key appearance/application of hypergraphs not mentioned so far. the erdos-rado sunflower lemmas[1], a key discovery of extremal graph/set theory, are about an intrinsic order or emergent "structure" to "random" hypergraphs if certain somewhat modest constraints are satisfied. this lemma shows up in numerous important lower bounds proofs in monotone circuit theory in computer science, including new versions that strengthen or generalize the lemma.[2]

because of their particular role in these "bottleneck"-type proofs its not outlandish to conjecture that variations might be crucial in some future-established comp sci complexity class separations.

[1] erdos-rado sunflowers survey/refs, TCS.se

[2] The Monotone Complexity of k-Clique on Random Graphs by Rossman, containing new stronger lemmas on "quasi sunflowers"

[3] Razborovs method of approximations by WT Gowers

probably many thms and applications of math that dont explicitly refer to hypergraphs are actually related to them implicitly & could be recast in those terms. because hypergraphs are equivalently just "sets of sets". in this way they're also often interchangeable with/analogous to a 2d boolean array in computer science (and how ubiquitous is that structure in both software and mathematics? in computer science it might be referred to as a "design pattern" or even just a simple "discrete structure").

here is one key appearance/application of hypergraphs not mentioned so far. the erdos-rado sunflower lemmas[1], a key discovery of extremal graph/set theory, are about an intrinsic order or emergent "structure" to "random" hypergraphs if certain somewhat modest constraints are satisfied. this lemma shows up in numerous important lower bounds proofs in monotone circuit theory in computer science, including new versions that strengthen or generalize the lemma.[2]

because of their particular role in these "bottleneck"-type proofs its not outlandish to conjecture that variations might be crucial in some future-established comp sci complexity class separations.

[1] erdos-rado sunflowers survey/refs, TCS.se

[2] The Monotone Complexity of k-Clique on Random Graphs by Rossman, containing new stronger lemmas on "quasi sunflowers"

[3] Razborovs method of approximations by WT Gowers