Some extremal problems for 0-1 matrices
Speaker: Eoin Long
Definition 1. Let τ be an element of Sk . A permutation π ∈ Sn is said to
contain τ if there exists 1 ≤ i1 < · · · < ik ≤ n such that τ (r) < τ (s) iff
π(ir ) < π(is ). Otherwise we say that π avoids τ .
Theorem 2. (Stanley-Wilf Conjecture, late 80s) Let τ ∈ Sk be fixed. Then the
number of permutations avoiding τ is at most cn for some constant c = c(τ ).
The conjecture was proven in 2004 by Marcus and Tardos. In fact, they
proved slightly more, establishing an old conjecture of Füredi and Hajnal which
Klazar had earlier shown implies the Stanley-Wilf conjecture. Before stating
this result we will need some definitions.
Definition 3. Let P be a fixed k × k 0-1 matrix. An n × n 0-1 matrix M is said
to contain P if there exists 1 ≤ i1 < · · · < ik ≤ n and 1 ≤ j1 < · · · < jk ≤ n s.t.
whenever Pr,s = 1 we also have Mir ,js = 1. Otherwise we say M avoids P .
Note that in this definition we do not insist that Mir ,js = 0 when Pi,j = 0.
Definition 4. Given an n × n 0-1 matrix M we write w(M ) for the number
of 1 entries in M . Given a fixed k × k 0-1 matrix P we write ex(n, P ) =
maxM 6⊂P w(M ).
Theorem 5. (Conjectured by Füredi-Hajnal, 1992) Given any permutation
matrix P , ex(n, P ) = O(n).
In the first half of this talk I hope to give a complete proof of the Marcus Tardos proof. In the second (time permitting) I hope to discuss a counterexample
constructed by Pach and Tardos to another conjecture of Füredi and Hajnal.
Definition 6. Given n × n 0-1 matrix M with rows {r1 , · · · , rn } and columns
{c1 , · · · , cn } let G(M ) denote the ordered graph with ordered vertex set
{r1 , · · · , rn , c1 , · · · , cn } in which ri cj is an edge of G(M ) iff Mi,j = 1. Denote
by G∗ (M ) be the corresponding unordered graph.
Conjecture (Füredi-Hajnal, 1992): Given any fixed k × k 0-1 matrix P we
have ex(n, P ) = O(exgraph (2n, G∗ (P ))log(n)).
Here exgraph (n, H) denotes the usual graph extremal number for H. In
other words the extremal number for a 0-1 n×n matrix to contain a fixed matrix
P is not too far from the extremal number for the graph G∗ (P ) corresponding
to P . In fact the conjecture fails a 4 × 4 matrix P with G∗ (P ) = C8 .
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