New developments in hypergraph Ramsey theory Dhruv Mubayi Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago IHP, Paris, January 30, 2018
Outline • Classical Ramsey numbers Generalizations and Extensions • Erd˝ os-Hajnal Problem • Erd˝ os-Rogers Problem • Erd˝ os-Gy´ arf´ as-Shelah Problem • A proof idea (Stepping up with zigzags)
Ramsey theory for hypergraphs Definition Given k ≥ 2 and k -uniform hypergraphs H 1 , H 2 , the ramsey number r ( H 1 , H 2 ) is the minimum N such that every red/blue coloring of the k -sets of [ N ] results in a red copy of H 1 or a blue copy of H 2 . Write r k ( s , n ) := r ( K k s , K k n ) .
Ramsey theory for hypergraphs Definition Given k ≥ 2 and k -uniform hypergraphs H 1 , H 2 , the ramsey number r ( H 1 , H 2 ) is the minimum N such that every red/blue coloring of the k -sets of [ N ] results in a red copy of H 1 or a blue copy of H 2 . Write r k ( s , n ) := r ( K k s , K k n ) . Observation Note that r k ( s , n ) ≤ N is equivalent to saying that every N-vertex K k s -free k-uniform hypergraph H has α ( H ) ≥ n.
Small examples Example Graphs: r 2 (3 , 3) = 6 r 2 (4 , 4) = 18 r 2 (3 , 3 , 3) = 17 Example Hypergraphs: r 3 (4 , 4) = 13 (McKay-Radziszowski 1991)
Graphs Theorem (Spencer 1977, Conlon 2008) √ 4 n 2 e n 2 n / 2 < r 2 ( n , n ) < (1 + o (1)) n c log n / log log n Theorem (Ajtai-Koml´ os-Szemer´ edi 1980, Kim 1995) � n 2 � r 2 (3 , n ) = Θ log n
Graphs Theorem (Spencer 1977, Conlon 2008) √ 4 n 2 e n 2 n / 2 < r 2 ( n , n ) < (1 + o (1)) n c log n / log log n Theorem (Ajtai-Koml´ os-Szemer´ edi 1980, Kim 1995) � n 2 � r 2 (3 , n ) = Θ log n Theorem For fixed s ≥ 3 n ( s +1) / 2+ o (1) < r 2 ( s , n ) < n s − 1+ o (1)
Hypergraphs - diagonal case Definition (tower function) twr i +1 ( x ) = 2 twr i ( x ) . twr 1 ( x ) = x and Theorem (Erd˝ os-Hajnal-Rado 1952/1965) 2 cn 2 < r 3 ( n , n ) < 2 2 n For fixed k ≥ 3 , twr k − 1 ( cn 2 ) < r k ( n , n ) < twr k ( c ′ n ) Conjecture (Erd˝ os $500) r 3 ( n , n ) > 2 2 cn .
An equivalent statement Definition P 5 is the ordered 4-uniform hypergraph with 5 vertices v 1 < v 2 < v 3 < v 4 < v 5 and two edges ( v 1 , v 2 , v 3 , v 4 ) and ( v 2 , v 3 , v 4 , v 5 ) . Theorem (M-Suk 2017) or 4 ( P 5 , n ) > 2 2 c ′ n . r 3 ( n , n ) > 2 2 cn ⇐ ⇒
Ordered tight path versus clique Definition A tight path of size s is an ordered hypergraph H, denoted by P k s with s vertices v 1 < · · · < v s ∈ [ n ] such that ( v j , v j +1 , . . . , v j + k − 1 ) is an edge for j = 1 , . . . , s − k + 1 . Let or k ( P s , n ) = or ( P ( k ) , K ( k ) ) . s n Theorem (M-Suk 2017) � � n n r 3 s − 3 , . . . , ≤ or 4 ( P s , n ) ≤ r 3 ( n , . . . , n ) . s − 3 � �� � � �� � s − 3 times s − 3 times
Hypergraphs - The off-diagonal conjecture Conjecture (Erd˝ os-Hajnal 1972) For fixed s > k ≥ 3 we have r k ( s , n ) > twr k − 1 ( cn ) . In particular, r k ( k + 1 , n ) > twr k − 1 ( cn ) . Theorem (Erd˝ os-Hajnal 1972) r 3 (4 , n ) > 2 cn . Consequently, the conjecture holds for k = 3 .
Hypergraphs - The off-diagonal conjecture Conjecture (Erd˝ os-Hajnal 1972) For fixed s > k ≥ 3 we have r k ( s , n ) > twr k − 1 ( cn ) . In particular, r k ( k + 1 , n ) > twr k − 1 ( cn ) . Theorem (Erd˝ os-Hajnal 1972) r 3 (4 , n ) > 2 cn . Consequently, the conjecture holds for k = 3 . Proof. Let T be a random graph tournament on N vertices and form a 3-uniform hypergraph by making each cyclically oriented triangle a hyperedge. There is no K (3) and yet the independence 4 number is n = O (log N ).
Hypergraphs - The off-diagonal conjecture Theorem (Erd˝ os-Hajnal) The conjecture holds for s = 2 k − 1 − k + 3 ; i.e., r 4 (7 , n ) > 2 2 cn . Theorem (Conlon-Fox-Sudakov 2009) The conjecture holds for s = ⌈ 5 k / 2 ⌉ − 3 . Theorem (M-Suk 2017, Conlon-Fox-Sudakov 2017) The conjecture holds for all s ≥ k + 3 .
Hypergraphs - The off-diagonal conjecture Theorem (Erd˝ os-Hajnal) The conjecture holds for s = 2 k − 1 − k + 3 ; i.e., r 4 (7 , n ) > 2 2 cn . Theorem (Conlon-Fox-Sudakov 2009) The conjecture holds for s = ⌈ 5 k / 2 ⌉ − 3 . Theorem (M-Suk 2017, Conlon-Fox-Sudakov 2017) The conjecture holds for all s ≥ k + 3 . The open cases are r 4 (5 , n ) and r 4 (6 , n ) and their k -uniform counterparts.
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ):
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972)
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017)
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017) • 2 n c log log n (M-Suk 2018?)
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017) • 2 n c log log n (M-Suk 2018?) • 2 n c log n (M-Suk 2018?)
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017) • 2 n c log log n (M-Suk 2018?) • 2 n c log n (M-Suk 2018?) Lower bounds for r 4 (6 , n ):
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017) • 2 n c log log n (M-Suk 2018?) • 2 n c log n (M-Suk 2018?) Lower bounds for r 4 (6 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972)
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017) • 2 n c log log n (M-Suk 2018?) • 2 n c log n (M-Suk 2018?) Lower bounds for r 4 (6 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 n c log n (M-Suk 2017)
r 4 (5 , n ) and r 4 (6 , n ) Lower bounds for r 4 (5 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 cn 2 (M-Suk 2017) • 2 n c log log n (M-Suk 2018?) • 2 n c log n (M-Suk 2018?) Lower bounds for r 4 (6 , n ): • 2 cn (implicit in Erd˝ os-Hajnal 1972) • 2 n c log n (M-Suk 2017) • 2 2 cn 1 / 5 (M-Suk 2018?)
The off-diagonal conjecture - almost solved Theorem (M-Suk 2018) r 4 (6 , n ) > 2 2 cn 1 / 5 r 4 (5 , n ) > 2 n c log n and for fixed k ≥ 4 r k ( k + 1 , n ) > twr k − 2 ( n c log n ) r k ( k + 2 , n ) > twr k − 1 ( cn 1 / 5 ) r k ( k + 1 , k + 1 , n ) > twr k − 1 ( cn ) .
Many Colors Theorem (Erd˝ os-Rado, Erd˝ os-Hajnal-Rado, Duke-Lefmann-R¨ odl, Axenovich-Gy´ arf´ as-Liu-M) For s > k ≥ 2 there are c and c ′ with ) < twr k ( c ′ q log q ) . twr k ( c q ) < r k ( s , . . . , s � �� � q times Special Case: (Erd˝ os-Szekeres 1935) 2 c q < r 2 (3 , . . . , 3 ) < 2 c ′ q log q . � �� � q times
The Erd˝ os-Hajnal Hypergraph Ramsey Problem Definition (Erd˝ os-Hajnal 1972) � s � For 1 ≤ t ≤ , let r k ( s , t ; n ) be the minimum N such that every k red/blue coloring of the k-sets of [ N ] results in an s-set that contains at least t red k-subsets or an n-set all of whose k-subsets are blue (i.e., a blue K k n ). Example � � s � � r k s , ; n = r k ( s , n ) k
The Erd˝ os-Hajnal Hypergraph Ramsey Problem Problem (Erd˝ os-Hajnal 1972) � s � , there is a well-defined value t 1 = h ( k ) As t grows from 1 to 1 ( s ) k at which r k ( s , t 1 − 1; n ) is polynomial in n while r k ( s , t 1 ; n ) is exponential in a power of n, another well-defined value t 2 = h ( k ) 2 ( s ) at which it changes from exponential to double exponential in a power of n and so on, and finally a well-defined � s � value t k − 2 = h ( k ) k − 2 ( s ) < at which it changes from twr k − 2 to k twr k − 1 in a power of n.
The Erd˝ os-Hajnal Hypergraph Ramsey Conjectures Conjecture (Erd˝ os-Hajnal $500) The first jump h ( k ) 1 ( s ) is one more than the number of edges in the k-uniform hypergraph obtained from a complete k-partite k-uniform hypergraph on s vertices with almost equal part sizes, by repeating this construction recursively within each part. Conjecture (Erd˝ os-Hajnal) h ( k ) r k ( k + 1 , t ; n ) = twr t − 1 ( n Θ ( 1 ) ) . ( k + 1) = i + 2 ⇐ ⇒ i Theorem (Erd˝ os-Hajnal) 2 cn < r k ( k + 1 , t ; n ) < twr t − 1 ( n c ′ ) .
Stepping up Conjecture (Erd˝ os-Hajnal 1972) r k ( k + 1 , t ; n ) = twr t − 1 ( n Θ ( 1 ) ) . Theorem (M-Suk 2018) For fixed 3 ≤ t ≤ k − 2 , � twr t − 1 ( n k − t +1+ o (1) ) twr t − 1 ( n k − t +1+ o (1) ) > r k ( k +1 , t ; n ) > twr t − 1 ( n ( k − t +1) / 2+ o (1) ) where the first inequality is when k − t is even and the second when k − t is odd.
The Erd˝ os-Rogers Problem Definition A t-independent set in a k-uniform hypergraph H is a vertex subset that contains no K k t . When t = k it is just an independent set. Write α t ( H ) for the maximum size of a t-independent set in H. Definition (Erd˝ os-Rogers function 1962) f k t , s ( N ) = min { α t ( H ) : | V ( H ) | = N , K k s �⊂ H } . Example f 2 2 , 3 ( N ) < n ⇐ ⇒ ∃ K 3 -free G with N vertices and α ( G ) < n. r k ( s , n ) = min { N : f k k , s ( N ) ≥ n } .
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