Title: A Refutation of the Pach-Tardos Conjecture for 0-1 Matrices
Domain: Computer Science

Question: Refute the Pach-Tardos conjecture for 0-1 matrices.

Background: ## Forbidden 0--1 Matrix Theory | - **Origins and Applications**: The theory of forbidden 0--1 matrices emerged in the late 1980s, applied to problems in discrete and computational geometry by Mitchell, Pach and Sharir, and Füredi. It generalizes Davenport-Schinzel theory and Turán theory to ordered bipartite graphs, influencing areas like computational geometry, data structure analysis, and the development of the graph parameter "twin width". | - **Definitions**: A matrix $A \in \{0,1\}^{n \times m}$ contains a pattern $P \in \{0,1\}^{k \times l}$ if $P$ can be obtained from $A$ by removing rows/columns and flipping 1s to 0s. If not, $A$ is $P$-free. The extremal function $\text{Ex}(P, n, m)$ is the maximum weight (number of 1s) of a $P$-free matrix in $\{0,1\}^{n \times m}$, with $\text{Ex}(P, n) = \text{Ex}(P, n, n)$. $P$ can be seen as the adjacency matrix of an ordered bipartite graph. Turán's extremal function $\text{ExT}(H, n)$ is the maximum number of edges in an $n$-vertex graph without $H$ as a subgraph. | | ## Pattern Classification | - **Turán's Extremal Function Classification**: Forbidden subgraphs are classified based on Turán's extremal function: | - Non-bipartite $H$: $\text{ExT}(H, n) = \Theta(n^2)$. | - Bipartite $H$ with a cycle: $\text{ExT}(H, n) = \Omega(n^{1+c_1})$ and $O(n^{1+c_2})$ for $0 < c_1 < c_2 < 1$. | - Acyclic (forest) $H$: $\text{ExT}(H, n) = \Theta(n)$. | - **0--1 Matrix Patterns**: Füredi and Hajnal noted $\text{Ex}(P, n) = \Omega(\text{ExT}(G(P), 2n))$ and found examples where $\text{Ex}(P, n) = \omega(\text{ExT}(G(P), 2n))$, such as $P_1$ and $Q_3$, which are acyclic matrices but have $\text{Ex}(P_1, n) = \Theta(n \log n)$ and $\text{Ex}(Q_3, n) = \Theta(n \alpha(n))$. | | ## Füredi-Hajnal and Pach-Tardos Conjectures | - **Füredi-Hajnal Conjectures**: | - **Permutation Matrix Conjecture (conj:FH-perm)**: If $P$ is a permutation matrix, then $\text{Ex}(P, n) = O(\text{ExT}(G(P), n)) = O(n)$. | - **Logarithmic Factor Conjecture (conj:FH-logn)**: For any $P$, $\text{Ex}(P, n) = O(\text{ExT}(G(P), n) \cdot \log n)$. | - **Acyclic Pattern Conjecture (conj:FH-acyclic)**: For any acyclic $P$, $\text{Ex}(P, n) = O(\text{ExT}(G(P), n) \cdot \log n) = O(n \log n)$. | - **Pach-Tardos Conjecture (conj:PachTardos)**: For an acyclic 0--1 pattern $P$: | - **Weak Version**: $\text{Ex}(P, n) = O(n \log^{C_P} n)$ for some constant $C_P$. | - **Strong Version**: $\text{Ex}(P, n) = O(n \log^{\|P\|_1 - 3} n)$.

Definitions: ## Covering Pattern | A pattern \( M \in \{0,1\}^{\alpha \times \beta} \) with rows indexed by \( 0 \leq i < \alpha \) and columns by \( 0 \leq j < \beta \). There exists a distinguished row \( k^* \) such that: | 1. \( M(k^*, 0) = M(k^*, \beta - 1) = 1 \). | 2. Let \( J \) be the set of row indices (excluding \( k^* \)) with at least two 1s. \( J \) contains at most one element \( j_0 < k^* \). For any \( j \in J \) where \( j > k^* \), some column contains two 1s in rows \([k^*, j]\). | 3. For \( l \in J \), define \( \text{first}(l) \) and \( \text{last}(l) \) as the column indices of the first and last 1s in row \( l \). The real intervals \([\text{first}(l), \text{last}(l)]\) cover the entire range \([0, \beta - 1]\). | | ## Landmark Columns | For a 1 in matrix \( A \) at position \( (r, c) \): | - **F** and **L**: The first and last 1s in row \( r \) following column \( c \). | - **a_j** and **b_j**: Consecutive 1s in row \( r \) such that \( F \leq a_j < b_j \leq a_{j+1} \) and \( b_j - a_j \) is maximized. | | ## Signatures | For a 1 in matrix \( A \) at position \( (r, c) \) that is not one of the last two 1s in its row: | - **sig0(r, c)**: A vector \( \left( \left\lfloor \log_\zeta (F - c) \right\rfloor, \left\lfloor \log_\zeta (b_1 - a_1) \right\rfloor, \ldots, \left\lfloor \log_\zeta (b_{t-1} - a_{t-1}) \right\rfloor \right) \). | - **sig1(r, c)**: A vector \( \left( \left\lfloor \log_{1+\epsilon} (a_1 - c) \right\rfloor, \ldots, \left\lfloor \log_{1+\epsilon} (a_{t-1} - c) \right\rfloor \right) \) and the relative positions of \( \left\lfloor \log_{1+\epsilon} (b_j - c) \right\rfloor \) to the vector elements. | - **sig2(r, c)**: A vector \( \left( \left\lfloor \log_{1+\epsilon} (b_1 - c) \right\rfloor, \ldots, \left\lfloor \log_{1+\epsilon} (b_{t-1} - c) \right\rfloor \right) \), the relative positions of \( \left\lfloor \log_{1+\epsilon} (a_j - c) \right\rfloor \) to the vector elements, and a vector \( \left( \min \left\{ \frac{a_2 - b_1}{(b_1 - c)/2}, 3t \right\}, \ldots, \min \left\{ \frac{L - b_{t-1}}{(b_{t-1} - c)/2}, 3t \right\} \right) \).

Methodology: ## 1. Matrix Construction | - **Construct Special Matrices**：Define a new class of 0-1 matrices \( A[b, m] \) with rows indexed by \([m] \times [m]^b\) and columns by \([m]^b \times \{0, 1\}^b\). Fill the elements with \( A((s, r), (c, i)) = 1 \) if \( r = c + s \cdot i \). | - **Introduce Alternating Coordinate Offsets**：Based on \( A[b, m] \), construct \( A_t[b, m] \) matrices by restricting columns (e.g., to those with weight \( t \)). | | ## 2. Refuting the Conjecture | - **Construct Counterexample Matrices**：Using the constructed \( A[b, m] \) matrices, demonstrate that for two weight-6 patterns \( S_0 \) and \( S_1 \), their extremal functions have a lower bound of \( n2^{\Omega(\sqrt{\log n})} \). | - **Expand the Scope of Counterexamples**：Further prove that all patterns in the covering pattern class have similar lower bounds. This extends the counterexamples from \( S_0 \) and \( S_1 \) to a broader covering pattern category. | | ## 3. Alternating Pattern Class Analysis | - **Lower Bound Proof**：Using the constructed \( A_t[b, m] \) matrices, show that the extremal function of the alternating pattern class \( P_t \) has a lower bound of \( \Omega(n(\log n / \log \log n)^t) \). | - **Upper Bound Proof**：Through an analysis and marking process, demonstrate that the extremal function of the alternating pattern class \( P_t \) has an upper bound of \( O(n(\log n / \log \log n)^t) \).

Answer: Refuting the Pach–Tardos Conjecture for 0–1 Matrices | | The Pach–Tardos conjecture predicted an upper bound on the extremal function Ex(M, n)—the maximum number of 1s in an n × n 0–1 matrix that avoids a given forbidden submatrix M. | The work here constructs explicit 0–1 matrices with far more 1s than the conjecture would allow, thereby disproving it for certain patterns. | | ⸻ | | Core Construction | | We define a base matrix A[b, m]: | • Rows: indexed by [m] × [m]^b | • Columns: indexed by [m]^b × {0,1}^b | • Entry rule: | | A((s, r), (c, i)) = 1 if r = c + s · i | 0 otherwise | | | • Arithmetic is done coordinatewise over integers, with i treated as a 0–1 indicator vector. | | Key property (Lemma 2): | Each column has on average at least m/(b+1) - 1 ones, so | ‖A[b, m]‖₁ ≥ Ω(m/b) × (#columns). | | ⸻ | | Pattern Avoidance | | The forbidden patterns of interest are: | • S₀ and S₁: small fixed 0–1 matrices that arise in the conjecture. | • Covering patterns: defined to have a “distinguished” row with 1s at the extremes, plus certain row–interval covering properties. | | Lemma 4 gives structural constraints on the positions of 1s in A[b, m], relating the order of row/column indices to coordinate differences in the [m]^b part. | | ⸻ | | Main Lower Bound Result | | Theorem 1 | For S₀ and S₁, | | Ex(S₀, n), Ex(S₁, n) ≥ n² · √(log n − O(log log n)) | | Reasoning: | 1. Show A[b, m] contains no S₀ or S₁: | • Assume S₀ is embedded in A. | • Define difference vectors x, y, z from certain column pairs. | • Lemma 4 forces their first nonzero coordinates to be incompatible with the relative order constraints implied by S₀. | • Same argument works for S₁ (swapping two rows). | 2. Set m = 2ᵇ so that A is square: n = 2^(b² + b). | 3. With b ≈ √(log n), the density gives the claimed bound. | | Thus, these matrices are both S₀- and S₁-free yet have Ω(n² √log n) ones, contradicting the conjecture. | | ⸻ | | Extension to Covering Patterns | | Theorem 2 | If M is any covering pattern (per Definition 1), then A[b, m] is M-free and | | Ex(M, n) ≥ n² · √(log n − O(log log n)) | | Proof uses the same difference–vector argument: the covering property forces a sum of certain differences to match a global difference, but Lemma 4 forbids this. | | ⸻ | | Polylogarithmic Lower Bounds for Alternating Patterns | | For alternating patterns Pₜ, modify the construction to Aₜ[b, m] by restricting to columns with exactly t ones in their {0,1}^b part and offsetting even/odd ones differently. | | Theorem 3 | For fixed t ≥ 1: | | Ex(Pₜ, n) = Ω( n · (log n / log log n)ᵗ ) | | The proof: | • Assume a Pₜ instance appears in Aₜ. | • Label rows/columns and examine the positions of first nonzero coordinates in column differences. | • Monotonicity and parity constraints force a coordinate–value conflict, so Pₜ cannot occur. | | ⸻ | | Final Implication | | By explicitly constructing large, pattern-free matrices for S₀, S₁, and more general covering patterns, we get extremal functions far above the Pach–Tardos conjecture’s prediction. This refutes the conjecture for these patterns, replacing its proposed bound with much larger, explicit lower bounds.

Checklist: 1. **Correct Definition of Forbidden Patterns**  
   Check that forbidden patterns in 0–1 matrices are correctly defined:  
   - A pattern P is contained in a matrix A if A can be transformed into P by deleting rows/columns and flipping some 1s to 0s.  
   - Extremal function Ex(P, n) is defined as the maximum number of 1s in an n×n matrix avoiding P.  
   - The connection to bipartite graphs (rows/columns correspond to bipartition, order matters) is clearly stated.  

2. **Accurate Definition of Matrix Construction**  
   Verify the construction of the base matrices A[b, m] and their variant At[b, m]:  
   - Rows indexed by [m] × [m]^b, columns by [m]^b × {0,1}^b (or restricted to Hamming weight t for At).  
   - Entry rule: A((s,r),(c,i)) = 1 if r = c + s·i (for A[b,m]); for At[b,m], offsets split into even/odd coordinates, using s and (m+1−s).  
   - Ensure lexicographic ordering and coordinate-wise arithmetic are correctly specified.  

3. **Valid Introduction of Alternating Coordinate Offsets**  
   Confirm that in the At[b,m] construction, introducing alternating offsets (i_even, i_odd) is valid and necessary:  
   - i_even contains the 2nd, 4th, … 1s of i; i_odd contains the 1st, 3rd, … 1s.  
   - Offsets s and m+1−s are applied accordingly.  
   - This modification ensures Pt-free property (alternating patterns cannot appear).  

4. **Exact Statement and Proof of Lemma 2**  
   Verify Lemma 2: For the base construction A[b, m], the number of ones satisfies  
   \[
   \|A[b,m]\|_1 \;\ge\; 2^b m^b \cdot (m/(b+1) - 1),
   \]  
   i.e. each column contains on average at least m/(b+1) − 1 ones, so the matrix has Ω(m/b) times as many ones as columns. Confirm proof reasoning is correct (expectation over columns, non-overflow condition).  

5. **Proper Definition of Covering Patterns**  
   Ensure Definition 1 is correct:  
   - A covering pattern has a distinguished row with 1s in the first and last columns.  
   - Other rows with ≥2 ones satisfy interval covering constraints relative to the distinguished row.  
   - The union of these row intervals covers the whole column range.  

6. **Valid Proof of Theorem 2.1**  
   Confirm Theorem 2.1: For any covering pattern M, the constructed A[b, m] avoids M (M-free), and  
   \[
   Ex(M, n) \;\ge\; n^2 \cdot \sqrt{\log n - O(\log \log n)}.
   \]  
   Proof uses Lemma 4 and contradiction with interval covering property.  

7. **Exact Statement and Proof of Theorem 2.3**  
   Verify Theorem 2.3: For any fixed t ≥ 1, alternating pattern Pt has extremal function  
   \[
   Ex(P_t, n) = \Omega\big( n \cdot ( \tfrac{\log n}{\log \log n} )^t \big).
   \]  
   Confirm that At[b,m] is Pt-free, and density argument yields the bound.  

8. **Correct Definition of Landmark Columns and Signatures**  
   Ensure that landmark columns (F, L, a_j, b_j) are defined correctly as structural markers in each row, and that signatures (sig0, sig1, sig2) are constructed to encode distances and relative positions of landmarks.  

9. **Valid Proof of Lemma 6**  
   Confirm Lemma 6: If an entry A(r, c*) = 1 is not the last 1 of its sig0-type, then at least one of inequalities (I.1)–(I.t) holds (relating distances between landmarks and ζ). Proof checks intervals, applies log-binned comparisons, and establishes inequality bounds.  

10. **Valid Proof of Theorem 3.1**  
   Verify Theorem 3.1: For t ≥ 2, the extremal function of alternating patterns satisfies  
   \[
   Ex(P_t, n) = O\big( n \cdot ( \tfrac{\log n}{\log \log n} )^t \big).
   \]  
   Confirm that the 4-step marking process (using sig0, sig1, sig2) eliminates all unmarked 1s unless a Pt instance exists, bounding the number of 1s in Pt-free matrices.