The oriented Turán number of a given oriented graph F, denoted by (n,F), is the largest number of arcs in n-vertex F-free oriented graphs. This concept could be seen as an oriented version of the classical Turán number. In this paper, we first prove several propositions that give exact results for several oriented graphs. In particular, we determine all exact values of (n,F) for every oriented graph F with at most three arcs and sufficiently large n. After that, we prove a stability result and use it to determine the Turán number of an orientation of C_4. Finally, we prove oriented versions of the random zooming theorem by Fernández, Hyde, Liu, Pikhurko and Wu and the almost regular subgraph theorem by Erdős and Simonovits, and use them to obtain an oriented version of the Füredi-Alon-Krivelevich-Sudakov Theorem, which generalizes the famous KST Theorem.

For fixed graphs H and F, the generalized Turán number ex(n,H,F) is the maximum possible number of copies of a subgraph H in an n-vertex F-free graph. This article is a survey of this extremal function whose study was initiated in an influential 2016 article by Alon and Shikhelman (J. Combin. Theory, B, 121, 2016).

We say that a hypergraph $\mathcal{H}$ contains a graph $H$ as a trace if there exists some set $S\subset V(\mathcal{H})$ such that $\mathcal{H}|_S=\{h\cap S: h\in E(\mathcal{H})\}$ contains a subhypergraph isomorphic to $H$. We study the largest number of hyperedges in uniform hypergraphs avoiding some graph $F$ as trace. In particular, we determine this number in the case $F=K_3$ and any uniformity, resolving a special case of a conjecture of Mubayi and Zhao, and we improve a bound given by Luo and Spiro in the case $F=C_4$ and uniformity 3.

The Kneser cube Kn_n has vertex set 2^[n] and two vertices F,F' are joined by an edge if and only if F∩ F'=∅. For a fixed graph G, we are interested in the most number vex(n,G) of vertices of Kn_n that span a G-free subgraph in Kn_n. We show that the asymptotics of vex(n,G) is (1+o(1))2^n-1 for bipartite G and (1-o(1))2^n for graphs with chromatic number at least 3. We also obtain results on the order of magnitude of 2^n-1-vex(n,G) and 2^n-vex(n,G) in these two cases. In the case of bipartite G, we relate this problem to instances of the forbidden subposet problem.

In this paper, we study extremal and stability problems for Berge Hamiltonian cycles in r-uniform hypergraphs under a minimum degree condition. Let g_r(n,t)=n-tr+ttr-1, and let t=t(k) be the unique integer satisfying t-1r-1<k≤tr-1. Using a sharp Pósa-type degree sequence theorem of Salia, we prove an extremal upper bound on the number of hyperedges in an n-vertex r-uniform hypergraph with minimum degree at least k and with no Berge Hamiltonian cycle. We also prove a stability theorem in the dense range before the first minimizer of g_r(n,t): every near-extremal example is contained in one of two natural non-Hamiltonian constructions.

Given a hypergraph F, what is the largest chromatic number that an F-free hypergraph can have? In the case of graphs, this question is easy to answer: the chromatic number is unbounded if F contains a cycle, and the largest chromatic number of F-free graphs is k-1 if F is a forest on k vertices. The situation is more complicated for hypergraphs. The strong coloring of a hypergraph is a coloring of the vertices such that every hyperedge is rainbow. The weak coloring of a hypergraph is a coloring of the vertices such that no hyperedge is monochromatic. The strong/weak chromatic number of a hypergraph is the minimum number of colors in a strong/weak coloring of the hypergraph. Our question has been completely answered for the weak chromatic number, similarly to the graph case. We characterize the hypergraphs F such that F-free hypergraphs have bounded strong chromatic number. The only remaining case is when F is the 3-uniform expansion S_k^+ of a star with k edges. Concerning the strong chromatic number of S_k^+-free hypergraphs, we give bounds that are asymptitically sharp as k→∞. We also consider the same problem when the Berge copies of a graph F are forbidden. We characterize when the strong/weak chromatic numbers are bounded in this case, and obtain sharp results or bounds for specific trees. In particular, when F is a path, we give a tight bound when r=3 and an asymptotically sharp bound when r=4.

A graph G is uniquely H-saturated if it contains no copy of a graph Has a subgraph, but adding any new edge into G creates exactly one copy of H. Let C-4(+) be the diamond graph consisting of a 4-cycle C-4 with one chord and C-3(& lowast;) be the graph consisting of a triangle with a pendant edge. In this paper we prove that a nontrivial uniquely C-4(+)-saturated graph G has girth 3 or 4. Further, G has girth 4 if and only if it is a strongly regular graph with special parameters. For n > 18k(2)-24k+ 10 with k >= 2, there are no uniquely C-4(+)-saturated graphs on n vertices with k triangles. In particular, C3 & lowast; is the only nontrivial uniquely C-4(+) saturated graph with one triangle, and there are no uniquelyC(4)(+)saturated graphs with two, three or four triangles. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study the number of queries needed to identify a monotone Boolean function f:{0,1}^n →{0,1}. A query consists of a 0-1-sequence, and the answer is the value of f on that sequence. It is well-known that the number of queries needed is n⌊ n/2⌋+n⌊ n/2⌋+1 in general. Here we study a variant where f has k “reasons” to be 1, i.e., its disjunctive normal form has k conjunctions if the redundant conjunctions are deleted. This problem is equivalent to identifying an upfamily in 2^[n] that has exactly k minimal members. We find the asymptotics on the number of queries needed for fixed k. We also study the non-adaptive version of the problem, where the queries are asked at the same time, and determine the exact number of queries for most values of k and n.

We study the number of distance queries needed to identify certain properties of a hidden tree T on n vertices. A distance query consists of two vertices x, y, and the answer is the distance of x and y in T. We determine the number of queries an optimal adaptive algorithm needs to find two vertices of maximal distance up to an additive constant of at most 5, and the number of queries needed to identify the hidden tree asymptotically. We also study the non-adaptive versions of these problems, determining the number of queries needed exactly. (c) 2026 Published by Elsevier B.V.

Let ℋ be a hypergraph and F be a graph. If there exists a bijection between the hyperedges of ℋ and the edges of F such that each hyperedge contains its image, then we say that ℋ is a Berge copy of F, and the collection of Berge copies of F is denoted by Berge-F. Given r-graphs ℱ and ℋ, the generalized hyper-Turán number ex_r(n, ℋ, ℱ) is the maximum number of copies of ℋ in n-vertex ℱ-free r-graphs. We study ex_r(n, ℋ, Berge-F). For general ℋ, we connect this problem to counting copies of the shadow graph of ℋ in F-free graphs and obtain several exact results. In particular, we show that for any hypergraph ℋ, if k is sufficiently large, then ex_r(n, ℋ, Berge-K_k) is achieved by the balanced complete (k-1)-partite r-graph, generalizing a result of Morrison, Nir, Norin, Rzażewski and Wesolek [Journal of Combinatorial Theory, Series B, 162 (2023) 231–243] to the case of hypergraphs. We show that ex_r(n,K_s^r,Berge-F)≤ex_s(n,Berge-F) and present sufficient conditions for equality. We also consider the connected generalized Turán number for Berge paths.

We say that a set system ℱ is k-completely hyperseparating if for any vertex v, there are at most k sets in ℱ with intersection {v}. We determine the minimum size of such set systems on an n-element underlying set, generalizing a very recent result for k=2 by Batíková, Kepka, and Nemĕc. We say that ℱ is k-hyperseparating if for any vertex v, there are at most k sets in ℱ such that no other vertex is contained by exactly the same sets out of these k sets. We determine the minimum size of 2-hyperseparating set systems on an n-element underlying set.

The abstract chromatic number was introduced by Razborov and Coregliano in 2020 in using the language of model theory, and was used to extend the Erdős-Stone-Simonovits theorem to graphs with extra structures. A purely combinatorial version was introduced by Gerbner, Hama Karim and Kucheriya in 2026, who also showed that in addition to the asymptotic bound on the Turán number, the abstract chromatic number determines the asymptotics of several other Turán-type functions. We observe that the chromatic number is used here due to its special role in determining the asymptotics of the Turán number. For other extremal functions, other graph parameters may play a similar role and let us extend results in a similar fashion. We prove the appropriate generalizations and show two examples where this happens.

A Berge path of length k in an r-uniform hypergraph is a collection of k hyperedges h_1,…,h_k and k+1 vertices v_1,…,v_k+1 such that v_i, v_i+1∈ h_i for each 1≤ i≤ k. Győri, Katona and Lemons [European J. Combin. 58 (2016) 238–246] generalized the Erdős-Gallai theorem to Berge paths and established bounds for the Turán number of Berge paths. However, these bounds are sharp only when some divisibility conditions hold. Győri, Lemons, Salia and Zamora [J. Combin. Theory Ser. B 148 (2021) 239–250] determined the exact value of the Turán number of Berge paths in the case k≤ r. In this paper, we settle the final open case k>r, thereby completing the determination of the Turán number of Berge paths.

For a given graph F, the r-uniform suspension of F is the r-uniform hypergraph obtained from F by taking r-2 new vertices and adding them to every edge. In this paper, we consider Turán problems on suspension hypergraphs, and we obtain several general and exact results.

Given a graph F, a Berge copy of F (Berge-F for short) is a hypergraph obtained by enlarging the edges arbitrarily. Győri, Salia and Zamora [European J. Combin. 96 (2021) 103353] determined the maximum number of hyperedges in a connected r-uniform hypergraph on n vertices containing no Berge path of length k-1 for k≥ 2r+14 and sufficiently large n, and asked for the minimum k_0 such that this extremal number holds for all k≥ k_0. In this paper, we prove that the extremal number holds for all k≥ 2r+2 and fails for k≤ 2r+1, thereby completely resolving the problem posed by Gyori, Salia and Zamora. Moreover, we also improve the result of Füredi, Kostochka and Luo [Electron. J. Comb. 26(4) (2019) 4–31], who determined the maximum number of hyperedges in a 2-connected n-vertex r-uniform hypergraph containing no Berge cycle of length at least k for k≥ 4r and sufficiently large n, by showing that this extremal number holds for all k≥ 2r+2 and fails for k≤ 2r+1. Our approach reduces the Berge-Turán problem to a graph extremal problem, and applies recent work of Ai, Lei, Ning and Shi [Canad. J. Math. (2025) 1–27] on the feasibility of graph parameters and the Kelmans operation.

Let ℱ_5 denote the 3-uniform hypergraph on the vertex set {f_1,f_2,…,f_5} with hyperedges {f_1f_2f_3,f_1f_2f_4,f_3f_4f_5}. Recently, Balogh, Clemen and Luo determined the Turán number of a one-vertex blow-up of ℱ_5, more specifically, they blow up the vertex f_5 to t vertices, the resulting hypergraph is denoted by ℱ_5(f_5;t). They show that for infinitely many t, ℱ_5(f_5;t) has exponentially many extremal constructions and positive Turán density. In this paper, we determine the exact Turán number of the hypergraph obtained by blowing up f_3 of ℱ_5 to t vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Turán number of every blow-up of ℱ_5. For some special blow-ups of ℱ_5, for example, t-disjoint copies of ℱ_5, we determine the exact Turán number. We construct a hypergraph ℱ_sim(t) which is a subgraph of a blow-up of ℱ_5, and is contained in the hypergraph obtained by adding any new hyperedge to the Turán hypergraph (the balanced complete 3-partite hypergraph), but its extremal construction is not the Turán hypergraph. We also determine the exact Turán number of ℱ_sim(t).

In 2020, Coregliano and Razborov introduced a general framework to study limits of combinatorial objects, using logic and model theory. They introduced the abstract chromatic number and proved/reproved multiple Erdős-Stone-Simonovits-type theorems in different settings. In 2022, Coregliano extended this by showing that similar results hold when we count copies of K_t instead of edges. Our aim is threefold. First, we provide a purely combinatorial approach. Second, we extend their results by showing several other graph parameters and other settings where Erdős-Stone-Simonovits-type theorems follow. Third, we go beyond determining asymptotics and obtain corresponding stability, supersaturation, and sometimes even exact results.

For a family of graphs , a graph is called -free if it does not contain any member of as a subgraph. Given a collection of graphs (G_1,…,G_t) on the same vertex set V of size n, a rainbow graph on V is obtained by taking at most one edge from each G_i. We say that a collection is rainbow -free if it contains no rainbow copy of any member of . In this paper, we study the maximum values of min_i∈ [t]|E(G_i)|, ∑_i=1^t|E(G_i)| and ∏_i=1^t|E(G_i)| among rainbow {F,M_s+1}-free collections (G_1,…,G_t) on n vertices.

For a graph F, an r-uniform hypergraph H is a Berge-F if there is a bijection ϕ:E(F)→ E(H) such that e⊆ ϕ(e) for each e∈ E(F). Given a family ℱ of r-uniform hypergraphs, an r-uniform hypergraph is ℱ-free if it does not contain any member in ℱ as a subhypergraph. The Turán number of ℱ is the maximum number of hyperedges in an ℱ-free r-uniform hypergraph on n vertices. In this paper, some exact and general results on the Turán numbers for several types of Berge forests are obtained.

Given a positive integer r and a graph G with degree sequence d1, ... , dn, we define er(G) = & sum;ni=1 dri. We let exr(n, F) be the largest value of er(G) if G is an n-vertex F-free graph. We show that if F has a color-critical edge, then exr(n, F) = er(G) for a complete (chi(F) -1)-partite graph G (this was known for cliques and C5). We obtain exact results for several other non-bipartite graphs and also determine exr(n, C4) for r >= 3. We also give simple proofs of multiple known results. Our key observation is the connection to ex(n, Sr, F), which is the largest number of copies of Sr in n-vertex F-free graphs, where Sr is the star with r leaves. We explore this connection and apply methods from the study of ex(n, Sr, F) to prove our results. We also obtain several new results on ex(n, Sr, F). (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).