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ABSTRACT For $$B \subseteq \mathbb F_q^m$$, the nth affine extremal number of B is the maximum cardinality of a set $$A \subseteq \mathbb F_q^n$$ with no subset, which is affinely isomorphic to B. Furstenberg and Katznelson proved that for any $$B \subseteq \mathbb F_q^m$$, the nth affine extremal number of B is $o(q^n)$ as $$n \to \infty$$. By counting affine homomorphisms between subsets of $$\mathbb F_q^n$$, we derive new bounds and give new proofs of some previously known bounds for certain affine extremal numbers. At the same time, we establish corresponding supersaturation results. We connect these bounds to certain Ramsey-type numbers in vector spaces over finite fields. For $$s,t \geq 1$$, let $$R_q(s,t)$$ denote the minimum n such that in every red–blue coloring of the one-dimensional subspaces of $$\mathbb F_q^n$$, there is either a red s-dimensional subspace or a blue t-dimensional subspace of $$\mathbb F_q^n$$. The existence of these numbers is a special case of a well-known theorem of Graham, Leeb and Rothschild. We improve the best-known upper bounds on $$R_2(2,t)$$, $$R_3(2,t)$$, $$R_2(t,t)$$ and $$R_3(t,t)$$. 

Abstract We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex‐transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for in every ‐connected graph any two longest cycles intersect in at least vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph, which can be used to improve the best known bounds toward all the aforementioned conjectures: First, we show that every connected vertex‐transitive graph on vertices contains a cycle (and hence path) of length at least , improving on from DeVos [arXiv:2302:04255, 2023]. Second, we show that in every ‐connected graph with , any two longest cycles meet in at least vertices, improving on from Chen, Faudree, and Gould [J. Combin. Theory, Ser. B,72(1998) no. 1, 143–149]. Our proof combines combinatorial arguments, computer search, and linear programming. 

Abstract We show that for every integer and large , every properly edge‐colored graph on vertices with at least edges contains a rainbow subdivision of . This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs. 

Abstract A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $$r\in (1,2)$$ there exists a bipartite graph H such that $$\mathrm{ex}(n,H)=\Theta(n^r)$$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $$k\geq 2$$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $$k\geq 2$$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $$^{\prime}$$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $$^{\prime}$$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $$^{\prime}$$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ . 

Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC_2k, the cycle of length 2k, isO(n^1+1/k). Simonovits established a corresponding supersaturation result forC_2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn^1+1/kcontains at leastc(e(G)/v(G))^2kcopies ofC_2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.