Improved bounds for the sunflower lemma Kewen Wu Peking U UC - PowerPoint PPT Presentation

Main result Applications Proof overview Open problems Thanks Improved bounds for the sunflower lemma Kewen Wu Peking U → UC Berkeley Joint work with Ryan Alweiss Shachar Lovett Jiapeng Zhang Princeton UCSD Harvard → USC 1 / 18

Main result Applications Proof overview Open problems Thanks Definitions Definition ( w -set system and r -sunflower) A w -set system is a family of sets of size at most w . 2 / 18

Main result Applications Proof overview Open problems Thanks Definitions Definition ( w -set system and r -sunflower) A w -set system is a family of sets of size at most w . An r -sunflower is r sets S 1 , . . . , S r where Kernel : Y = S 1 ∩ · · · ∩ S r ; Petals : S 1 \ Y, . . . , S r \ Y are pairwise disjoint. 2 / 18

Main result Applications Proof overview Open problems Thanks Definitions Definition ( w -set system and r -sunflower) A w -set system is a family of sets of size at most w . An r -sunflower is r sets S 1 , . . . , S r where Kernel : Y = S 1 ∩ · · · ∩ S r ; Petals : S 1 \ Y, . . . , S r \ Y are pairwise disjoint. Example {{ 1 , 2 } , { 1 , 3 , 4 , 6 } , { 1 , 5 } , { 2 , 3 }} is a 4 -set system of size 4 . It has a 3 -sunflower {{ 1 , 2 } , { 1 , 3 , 4 , 6 } , { 1 , 5 }} with kernel { 1 } and petals { 2 } , { 3 , 4 , 6 } , { 5 } . 2 / 18

Main result Applications Proof overview Open problems Thanks Main result Theorem (Erd˝ os-Rado sunflower) Any w -set system of size s has an r -sunflower. 3 / 18

Main result Applications Proof overview Open problems Thanks Main result Theorem (Erd˝ os-Rado sunflower) Any w -set system of size s has an r -sunflower. Let’s focus on r = 3 . 3 / 18

Main result Applications Proof overview Open problems Thanks Main result Theorem (Erd˝ os-Rado sunflower) Any w -set system of size s has an r -sunflower. Let’s focus on r = 3 . os and Rado 1960: s = w ! · 2 w ≈ w w . Erd˝ 3 / 18

Main result Applications Proof overview Open problems Thanks Main result Theorem (Erd˝ os-Rado sunflower) Any w -set system of size s has an r -sunflower. Let’s focus on r = 3 . os and Rado 1960: s = w ! · 2 w ≈ w w . Erd˝ Kostochka 2000: s ≈ ( w log log log w/ log log w ) w . 3 / 18

Main result Applications Proof overview Open problems Thanks Main result Theorem (Erd˝ os-Rado sunflower) Any w -set system of size s has an r -sunflower. Let’s focus on r = 3 . os and Rado 1960: s = w ! · 2 w ≈ w w . Erd˝ Kostochka 2000: s ≈ ( w log log log w/ log log w ) w . Fukuyama 2018: s ≈ w 0 . 75 w . 3 / 18

Main result Applications Proof overview Open problems Thanks Main result Theorem (Erd˝ os-Rado sunflower) Any w -set system of size s has an r -sunflower. Let’s focus on r = 3 . os and Rado 1960: s = w ! · 2 w ≈ w w . Erd˝ Kostochka 2000: s ≈ ( w log log log w/ log log w ) w . Fukuyama 2018: s ≈ w 0 . 75 w . Now: s ≈ (log w ) w and this is tight for our approach. 3 / 18

Main result Applications Proof overview Open problems Thanks Actual bound and further refinement Theorem (Improved sunflower lemma) For some constant C , any w -set system of size s has an r -sunflower, where log w log log w + (log r ) 2 �� w . � Cr 2 · � s = 4 / 18

Main result Applications Proof overview Open problems Thanks Actual bound and further refinement Theorem (Improved sunflower lemma) For some constant C , any w -set system of size s has an r -sunflower, where log w log log w + (log r ) 2 �� w . � Cr 2 · � s = Recently, Anup Rao improved it to s = ( Cr (log w + log r ))) w . 4 / 18

Main result Applications Proof overview Open problems Thanks Applications – Theoretical computer science Circuit lower bounds Data structure lower bounds Matrix multiplication Pseudorandomness Cryptography Property testing Fixed parameter complexity Communication complexity ... 5 / 18

Main result Applications Proof overview Open problems Thanks Applications – Combinatorics Erd˝ os-Szemer´ edi sunflower lemma Intersecting set systems Packing Kneser graphs Alon-Jaeger-Tarsi nowhere-zero conjecture Thersholds in random graphs ... 6 / 18

Main result Applications Proof overview Open problems Thanks Section 3 Proof overview 7 / 18

Main result Applications Proof overview Open problems Thanks Make it robust Assume F = { S 1 , . . . , S m } is a w -set system. 8 / 18

Main result Applications Proof overview Open problems Thanks Make it robust Assume F = { S 1 , . . . , S m } is a w -set system. Define a width- w DNF f F as f F = � m � j ∈ S i x j . i =1 Example If F = {{ 1 , 2 } , { 1 , 3 , 4 , 6 } , { 1 , 5 } , { 2 , 3 }} , then f F = ( x 1 ∧ x 2 ) ∨ ( x 1 ∧ x 3 ∧ x 4 ∧ x 6 ) ∨ ( x 1 ∧ x 5 ) ∨ ( x 2 ∧ x 3 ) . 8 / 18

Main result Applications Proof overview Open problems Thanks Make it robust Assume F = { S 1 , . . . , S m } is a w -set system. Define a width- w DNF f F as f F = � m � j ∈ S i x j . i =1 Example If F = {{ 1 , 2 } , { 1 , 3 , 4 , 6 } , { 1 , 5 } , { 2 , 3 }} , then f F = ( x 1 ∧ x 2 ) ∨ ( x 1 ∧ x 3 ∧ x 4 ∧ x 6 ) ∨ ( x 1 ∧ x 5 ) ∨ ( x 2 ∧ x 3 ) . Definition (Satisfying system) F is satisfying if Pr [ f F ( x ) = 0] < 1 / 3 with Pr [ x i = 1] = 1 / 3 , 8 / 18

Main result Applications Proof overview Open problems Thanks Make it robust Assume F = { S 1 , . . . , S m } is a w -set system. Define a width- w DNF f F as f F = � m � j ∈ S i x j . i =1 Example If F = {{ 1 , 2 } , { 1 , 3 , 4 , 6 } , { 1 , 5 } , { 2 , 3 }} , then f F = ( x 1 ∧ x 2 ) ∨ ( x 1 ∧ x 3 ∧ x 4 ∧ x 6 ) ∨ ( x 1 ∧ x 5 ) ∨ ( x 2 ∧ x 3 ) . Definition (Satisfying system) F is satisfying if Pr [ f F ( x ) = 0] < 1 / 3 with Pr [ x i = 1] = 1 / 3 , i.e., Pr [ ∀ i ∈ [ m ] , S i �⊂ S ] < 1 / 3 with Pr [ x i ∈ S ] = 1 / 3 . 8 / 18

Main result Applications Proof overview Open problems Thanks Satisfyingness implies sunflower Assume F is a set system on ground set { 1 , . . . , n } . Lemma If F is satisfying, then it has 3 pairwise disjoint sets. 9 / 18

Main result Applications Proof overview Open problems Thanks Satisfyingness implies sunflower Assume F is a set system on ground set { 1 , . . . , n } . Lemma If F is satisfying, then it has 3 pairwise disjoint sets. 3 pairwise disjoint sets is a 3 -sunflower with empty kernel. 9 / 18

Main result Applications Proof overview Open problems Thanks Satisfyingness implies sunflower Assume F is a set system on ground set { 1 , . . . , n } . Lemma If F is satisfying, then it has 3 pairwise disjoint sets. 3 pairwise disjoint sets is a 3 -sunflower with empty kernel. Proof. Color x 1 , . . . , x n to red, green, blue uniformly and independenty. 9 / 18

Main result Applications Proof overview Open problems Thanks Satisfyingness implies sunflower Assume F is a set system on ground set { 1 , . . . , n } . Lemma If F is satisfying, then it has 3 pairwise disjoint sets. 3 pairwise disjoint sets is a 3 -sunflower with empty kernel. Proof. Color x 1 , . . . , x n to red, green, blue uniformly and independenty. By definition, F contains a purely red (green/blue) set w.p > 2 / 3 . 9 / 18

Main result Applications Proof overview Open problems Thanks Satisfyingness implies sunflower Assume F is a set system on ground set { 1 , . . . , n } . Lemma If F is satisfying, then it has 3 pairwise disjoint sets. 3 pairwise disjoint sets is a 3 -sunflower with empty kernel. Proof. Color x 1 , . . . , x n to red, green, blue uniformly and independenty. By definition, F contains a purely red (green/blue) set w.p > 2 / 3 . By union bound, F contains one purely red set, one purely green set, and one purely blue set w.p > 0 . 9 / 18

Main result Applications Proof overview Open problems Thanks Structure vs pseudorandomness Assume F = { S 1 , . . . , S m } , m > κ w is a w -set system. Define link F Y = { S i \ Y | Y ⊂ S i } , which is a ( w − | Y | ) -set system. Example If F = {{ 1 , 2 } , { 1 , 3 , 4 } , { 1 , 5 } , { 2 , 3 }} , then F { 2 } = {{ 1 } , { 3 }} . 10 / 18

Main result Applications Proof overview Open problems Thanks Structure vs pseudorandomness Assume F = { S 1 , . . . , S m } , m > κ w is a w -set system. Define link F Y = { S i \ Y | Y ⊂ S i } , which is a ( w − | Y | ) -set system. Example If F = {{ 1 , 2 } , { 1 , 3 , 4 } , { 1 , 5 } , { 2 , 3 }} , then F { 2 } = {{ 1 } , { 3 }} . If there exists Y such that |F Y | ≥ m/κ | Y | > κ w −| Y | , then we can apply induction and find an 3 -sunflower in F Y . 10 / 18

Main result Applications Proof overview Open problems Thanks Structure vs pseudorandomness Assume F = { S 1 , . . . , S m } , m > κ w is a w -set system. Define link F Y = { S i \ Y | Y ⊂ S i } , which is a ( w − | Y | ) -set system. Example If F = {{ 1 , 2 } , { 1 , 3 , 4 } , { 1 , 5 } , { 2 , 3 }} , then F { 2 } = {{ 1 } , { 3 }} . If there exists Y such that |F Y | ≥ m/κ | Y | > κ w −| Y | , then we can apply induction and find an 3 -sunflower in F Y . So induction starts at such F , that |F Y | <m/κ | Y | holds for any Y . Lemma Let κ ≥ (log w ) O (1) . If |F Y | < m/κ | Y | holds for any Y , then F is satisfying, which means F has 3 pairwise disjoint sets. 10 / 18

Main result Applications Proof overview Open problems Thanks Randomness preserves pseudorandomness Let F = { S 1 , . . . , S m } be a w -(multi-)set system. 11 / 18