Large homogeneous submatrices D aniel Kor andi EPFL July 17, - PowerPoint PPT Presentation

Large homogeneous submatrices D´ aniel Kor´ andi EPFL July 17, 2019 joint work with J´ anos Pach and Istv´ an Tomon

The general setup Let A be an n × n 0-1 matrix that does not contain some fixed P as a submatrix.

The general setup Let A be an n × n 0-1 matrix that does not contain some fixed P as a submatrix. Question What is the largest homogeneous (all-0 or all-1) submatrix?

The general setup Let A be an n × n 0-1 matrix that does not contain some fixed P as a submatrix. Question What is the largest homogeneous (all-0 or all-1) submatrix? ◮ Without forbidding P : c log n × c log n

The general setup Let A be an n × n 0-1 matrix that does not contain some fixed P as a submatrix. Question What is the largest homogeneous (all-0 or all-1) submatrix? ◮ Without forbidding P : c log n × c log n Is there a linear-size ( ε n × ε n ) homogeneous submatrix?

Bipartite graphs

Bipartite graphs Definition A bipartite graph is chordal if it has no induced cycle of length > 4.

Bipartite graphs Definition A bipartite graph is chordal if it has no induced cycle of length > 4.

Bipartite graphs Definition A bipartite graph is chordal if it has no induced cycle of length > 4.   1 1 1   ⇒ 0 1 1     1 0 1

Bipartite graphs Definition A bipartite graph is chordal if it has no induced cycle of length > 4.   1 1 1   ⇒ 0 1 1     1 0 1 adjacency matrix of such graph: totally balanced matrix

Bipartite graphs Definition A bipartite graph is chordal if it has no induced cycle of length > 4.   1 1 1   ⇒ 0 1 1     1 0 1 adjacency matrix of such graph: totally balanced matrix Theorem (Anstee-Farber et al., 1980s) A matrix is totally balanced iff its rows and columns can be � 1 1 � reordered so that there is no submatrix. 1 0

Bipartite graphs Definition A bipartite graph is chordal if it has no induced cycle of length > 4.   1 1 1   ⇒ 0 1 1     1 0 1 adjacency matrix of such graph: totally balanced matrix Theorem (Anstee-Farber et al., 1980s) A matrix is totally balanced iff its rows and columns can be � 1 1 � reordered so that there is no submatrix. 1 0 ⇒ There are linear-size subsets in both parts that together induce a complete or empty bipartite subgraph.

Points and directed lines

Points and directed lines Let n points and n directed lines be given in the plane.

Points and directed lines Let n points and n directed lines be given in the plane. � 1 if i ’th point is on the right of j ’th line Matrix: a ij = 0 if i ’th point is on the left of j ’th line

Points and directed lines Let n points and n directed lines be given in the plane.   1 1 0   ⇒ 1 1 1     0 0 1 � 1 if i ’th point is on the right of j ’th line Matrix: a ij = 0 if i ’th point is on the left of j ’th line

Points and directed lines Let n points and n directed lines be given in the plane.   1 1 0   ⇒ 1 1 1     0 0 1 � 1 if i ’th point is on the right of j ’th line Matrix: a ij = 0 if i ’th point is on the left of j ’th line Theorem (Keszegh-P´ alv¨ olgyi, 2019) � 1 0 � By reordering and inverting columns, one can get -free matrix. 0 1

Points and directed lines Let n points and n directed lines be given in the plane.   1 1 0   ⇒ 1 1 1     0 0 1 � 1 if i ’th point is on the right of j ’th line Matrix: a ij = 0 if i ’th point is on the left of j ’th line Theorem (Keszegh-P´ alv¨ olgyi, 2019) � 1 0 � By reordering and inverting columns, one can get -free matrix. 0 1 ⇒ There are ε n points and ε n lines such that either all points are on the right of all lines, or all on the left.

Continuous functions

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1]

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1] such that the f i are 1-intersecting,

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1] such that the f i are 1-intersecting, the g i are k -intersecting.

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1] such that the f i are 1-intersecting, the g i are k -intersecting. � 1 if f i ( x ) = g j ( x ) has a solution on [0 , 1] Matrix: a ij = 0 if f i ( x ) � = g j ( x ) on [0 , 1]

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1] such that the f i are 1-intersecting, the g i are k -intersecting.   1 1 ⇒   0 1 � 1 if f i ( x ) = g j ( x ) has a solution on [0 , 1] Matrix: a ij = 0 if f i ( x ) � = g j ( x ) on [0 , 1]

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1] such that the f i are 1-intersecting, the g i are k -intersecting.   1 1 ⇒   0 1 � 1 if f i ( x ) = g j ( x ) has a solution on [0 , 1] Matrix: a ij = 0 if f i ( x ) � = g j ( x ) on [0 , 1] Theorem (K-Pach-Tomon, 2019+) � 1 0 1 0 ··· 1 0 � Matrix is -free ( 2 k + 4 columns). 0 1 0 1 ··· 0 1

Continuous functions Two sets of continuous functions f 1 , . . . , f n and g 1 , . . . , g n on [0 , 1] such that the f i are 1-intersecting, the g i are k -intersecting.   1 1 ⇒   0 1 � 1 if f i ( x ) = g j ( x ) has a solution on [0 , 1] Matrix: a ij = 0 if f i ( x ) � = g j ( x ) on [0 , 1] Theorem (K-Pach-Tomon, 2019+) � 1 0 1 0 ··· 1 0 � Matrix is -free ( 2 k + 4 columns). 0 1 0 1 ··· 0 1 ⇒ One can select ε n of the f i and ε n of the g j so that either each equation f i = g j has a solution or none of them.

Does any P -free n × n matrix have an ε n × ε n homog submatrix?

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P   1 1 0 1   1 0 1 0 P :     0 0 1 1

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P   1 1 0 1   1 0 1 0 P :     0 0 1 1

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   1 1 0 1   1 0 1 0 P :     0 0 1 1

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   random matrix   1 1 0 1       1 0 1 0 P : A :         0 0 1 1  

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   random matrix   1 1 0 1 P [ a ij = 1] = n ε − 1       1 0 1 0 P : A :         0 0 1 1  

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   random matrix   1 1 0 1 P [ a ij = 1] = n ε − 1       1 0 1 0 P : A :     no large homog submx     0 0 1 1  

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   random matrix   1 1 0 1 P [ a ij = 1] = n ε − 1       1 0 1 0 P : A :     no large homog submx     0 0 1 1   o ( n ) copies of P

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P or P c ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   random matrix   1 1 0 1 P [ a ij = 1] = n ε − 1       1 0 1 0 P : A :     no large homog submx     0 0 1 1   o ( n ) copies of P

Does any P -free n × n matrix have an ε n × ε n homog submatrix? ◮ not necessarily, if there is a cycle in P or P c ◮ ∃ P -free matrix with no n 1 − ε × n 1 − ε homog submatrix   random matrix   1 1 0 1 P [ a ij = 1] = n ε − 1       1 0 1 0 P : A :     no large homog submx     0 0 1 1   o ( n ) copies of P P is simple if both P and P c are acyclic