Abstract The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width . The foremost open problem in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $$P\in \{0,1\}^{k\times l}$$ P ∈ { 0 , 1 } k × l is acyclic , meaning it is the bipartite incidence matrix of a forest, then $$\operatorname {Ex}(P,n) = O(n\log ^{C_P} n)$$ Ex ( P , n ) = O ( n log C P n ) , where $$\operatorname {Ex}(P,n)$$ Ex ( P , n ) is the maximum number of 1s in a P -free $$n\times n$$ n × n 0–1 matrix and $$C_P$$ C P is a constant depending only on P . This conjecture has been confirmed on many small patterns, specifically all P with weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $$\operatorname {Ex}(S_0,n),\operatorname {Ex}(S_1,n) \ge n2^{\Omega (\sqrt{\log n})}$$ Ex ( S 0 , n ) , Ex ( S 1 , n ) ≥ n 2 Ω ( log n ) , where $$S_0,S_1$$ S 0 , S 1 are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $$(P_t)$$ ( P t ) , specifically that for every $$t\ge 2$$ t ≥ 2 , $$\operatorname {Ex}(P_t,n)=\Theta (n(\log n/\log \log n)^t)$$ Ex ( P t , n ) = Θ ( n ( log n / log log n ) t ) . This is the first proof of an asymptotically sharp bound that is $$\omega (n\log n)$$ ω ( n log n ) .

Abstract In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph $$G_{n, p}$$ G n , p for $$p \geqslant C(\log n/n)^{1/\Delta }$$ p ⩾ C ( log n / n ) 1 / Δ , sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph H with $$\Delta (H) \leqslant \Delta$$ Δ ( H ) ⩽ Δ . However, this is typically only optimal when $$\Delta \in \{2,3\}$$ Δ ∈ { 2 , 3 } and H either contains a triangle ( $$\Delta = 2$$ Δ = 2 ) or many copies of $$K_4$$ K 4 ( $$\Delta = 3$$ Δ = 3 ). We go beyond this barrier for the first time and present a sparse blow-up lemma for cycles $$C_{2k-1}, C_{2k}$$ C 2 k - 1 , C 2 k , for all $$k \geqslant 2$$ k ⩾ 2 , and densities $$p \geqslant Cn^{-(k-1)/k}$$ p ⩾ C n - ( k - 1 ) / k , which is in a way best possible. As an application of our blow-up lemma we fully resolve a question of Nenadov and Škorić regarding resilience of cycle factors in sparse random graphs.

Abstract Let $$r \ge 3$$ r ≥ 3 be fixed and G be an n -vertex graph. A long-standing conjecture of Győri states that if $$e(G) = t_{r-1}(n) + k$$ e ( G ) = t r - 1 ( n ) + k , where $$t_{r-1}(n)$$ t r - 1 ( n ) denotes the number of edges of the Turán graph on n vertices and $$r - 1$$ r - 1 parts, then G has at least $$(2 - o(1))k/r$$ ( 2 - o ( 1 ) ) k / r edge disjoint r -cliques. We prove this conjecture.

Abstract We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite n -partite tournaments for $$n \in \{2,3,\ldots ,\omega \}$$ n ∈ { 2 , 3 , … , ω } , and for the class of finite semigeneric tournaments. We also prove that the generic $$\omega $$ ω -partite tournament and the generic semigeneric tournament have ample generics.