More than 40 years ago, Galvin, Rival and Sands showed that every K_s, s-free graph containing an n-vertex path must contain an induced path of length f(n), where f(n)→∞ as n→∞. Recently, it was shown by Duron, Esperet and Raymond that one can take f(n)=(loglog n)^1/5-o(1). In this note, we give a short self-contained proof that a K_s, s-free graphs with an n-vertex path contains an induced path of length at least (loglog n)^1-o(1). Combined with the recent remarkable example of Couëtoux, Defrain, and Raymond, which provides an upper bound of O((loglog n)^1+o(1)), this essentially resolves this old problem.

Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).

Let G be a Dirac graph, and let S be a vertex subset of G, chosen uniformly at random. How likely is the induced subgraph G[S] to be Hamiltonian? This question, proposed by Erdos and Faudree in 1996, was recently resolved by Draganic, Keevash, and Muyesser, in the setting of graphs. In this paper, we study a similar question for tournaments: If T is a tournament of high minimum degree, how likely is it for a random induced subtournament of T to be Hamiltonian? We prove an optimal bound on this probability, and extend the results to the regime where the subset is not sampled uniformly at random, but according to a p-biased measure.

Subdivisions of complete graphs play a central role in combinatorics, having deep connections to structural, extremal, and topological aspects of graph theory. A celebrated conjecture of Mader, proved independently by Bollobás and Thomason and by Komlós and Szemerédi, states that every graph of average degree of order h^2 contains a subdivision of K_h. In this paper, we consider the induced variant of this problem. A theorem of Kühn and Osthus implies that, for every fixed graph H and every s≥ 1, graphs of sufficiently large average degree contain either a copy of K_s,s or an induced subdivision of H. However, even for H=K_h, the best previous quantitative bounds were far from optimal. We prove nearly tight bounds for forcing induced subdivisions of K_h. We show that every K_s,t-free graph of average degree Ω_s,t(h^2(s-1)log^7(s-1) h) contains an induced subdivision of K_h, and that every C_2k-free graph with k ≥ 3 and average degree Ω_k(hlog^5 h) contains an induced subdivision of K_h. These bounds substantially improve the previously known results and are nearly optimal in both settings. They also hold if K_h is replaced by any other graph on h vertices.

The spherical random geometric graph G(n,d,p) is obtained by sampling n independent points uniformly on the unit sphere 𝕊^d-1⊆ℝ^d and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is p. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erdős–Rényi random graph G(n,p), as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and Rácz, who predicted that G(n,d,p) and G(n,p) are indistinguishable precisely when d ≫ n^3p^3(log p^-1)^3, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range 1/3 ≥ p ≥ n^-1/5polylog(n). The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that G(n,d,p) realizes a prescribed graph H: above the conjectured threshold, this probability is at most (1+o(1)) times the corresponding probability for G(n,p), with the signed triangle count of H appearing as the leading correction term.

Let w_1, …, w_m be positive real weights whose sum is 1, and let v_1, …, v_m be i.i.d. Bernoulli(p) random variables. If we let X=∑_i=1^m w_i v_i, then we conjecture that for all 0≤ p≤ 1/3 we have ℙ[X≥𝔼[X]]≥ p. In this short note, we observe a connection of this conjecture with a version of the Manickam-Miklós-Singhi conjecture, which allows one to prove it for sufficiently small values of p.

The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if $H$ is a bipartite graph, then an induced $H$-free and $K_{s,s}$-free graph cannot have much more edges than an $H$-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by $k$ on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erd\H{o}s-Hajnal conjecture, the problem of finding induced $C_4$-free subgraphs with large degree and bounding the average degree of $K_{s, s}$-free graphs which do not contain induced subdivisions of a fixed graph.

We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds.

Let $f(n, v, e)$ denote the maximum number of edges in a 3-uniform hypergraph on $n$ vertices which does not contain $v$ vertices spanning at least $e$ edges. A central problem in extremal combinatorics, famously posed by Brown, Erd\H{o}s and S\'os in 1973, asks whether $f(n, e+3, e)=o(n^2)$ for every $e \ge 3$. A classical result of S\'ark\"ozy and Selkow states that $f(n, e+\lfloor \log_2 e\rfloor+2, e)=o(n^{2})$ for every $e \ge 3$. This bound was recently improved by Conlon, Gishboliner, Levanzov and Shapira. Motivated by applications to other problems, Gowers and Long made the striking conjecture that $f(n, e+4, e)=O(n^{2-\varepsilon})$ for some $\varepsilon=\varepsilon(e)>0$. Conlon, Gishboliner, Levanzov and Shapira, and later, Shapira and Tyomkyn reiterated the following approximate version of this problem. What is the smallest $d(e)$ for which $f(n, e+d(e), e)=O(n^{2-\varepsilon})$ for some $\varepsilon=\varepsilon(e)>0$? In this paper, we prove that for each $e\geq 3$ we have $f(n, e+\lfloor \log_2 e\rfloor +38, e)=O(n^{2-\varepsilon})$ for some $\varepsilon>0$. This shows that one can already obtain power saving near the S\'ark\"ozy-Selkow bound at the cost of a small additive constant.

The Zarankiewicz problem, a cornerstone problem in extremal graph theory, asks for the maximum number of edges in an n-vertex graph that does not contain the complete bipartite graph K_s,s. While the problem remains widely open in the case of general graphs, the past two decades have seen significant progress on this problem for various restricted graph classes – particularly those arising from geometric settings – leading to a deeper understanding of their structure. In this paper, we develop a new structural tool for addressing Zarankiewicz-type problems. More specifically, we show that for any positive integer k, every graph with average degree d either contains an induced C_4-free subgraph with average degree at least k, or it contains a d-vertex subgraph with Ω_k(d^2) edges. As an application of this dichotomy, we propose a unified approach to a large number of Zarankiewicz-type problems in geometry, obtaining optimal bounds in each case.

Power saving for the Brown-Erdős-Sós problem, Discrete Analysis 2025:5, 16 pp. It has long been known that there are important connections between extremal questions concerning hypergraphs and extremal questions in additive combinatorics. This realization dates back at least as far as the famous $(6,3)$-theorem of Ruzsa and Szemerédi, which has several equivalent formulations, including the equally famous triangle removal lemma. In its hypergraph formulation, the theorem states that if a 3-uniform hypergraph has $n$ vertices and no six of those vertices span three edges, then there are $o(n^2)$ edges. (Here an "edge" means a triple of vertices that belongs to the hypergraph.) A moment's reflection shows that if the hypergraph is linear -- that is, no two edges overlap in more than one vertex -- then the only way that six vertices can span three edges is for those three edges to form what is sometimes called a "triforce", which is a configuration of the form $xyu, yzv, xzw$. Thus, another way to state the theorem is to say that for every $\delta>0$ there exists $n$ such that every linear hypergraph with $n$ vertices and at least $\delta n^2$ edges (note that the number of edges of a linear hypergraph is trivially at most $\binom n2$) contains a triforce. To see how this is relevant to additive combinatorics, let $G$ be an Abelian group of odd order $n$ and let $A$ be a subset of $G$ of density $\delta$. Form a 6-partite linear hypergraph $H$ with vertex sets $X,Y,Z,U,V,W$ that are each copies of $G$, and let $xyu$ be an edge if $y-x=u$ and $u\in A$, let $yzv$ be an edge if $z-y=v$ and $v\in A$, and let $xzw$ be an edge if $z-x=2w$ and $w\in A$. Then $H$ has positive density (independent of $n$) so for large enough $n$ it must contain a triforce. It is a small exercise to check that that triforce must contain exactly one vertex in each vertex set, which therefore gives us three elements $x,y,z$ of $G$ such that $y-x$ and $z-y$ belong to $A$ and $z-x$ belongs to $2.A$ (the dilate of $A$ by a factor of 2). From this we obtain $u,v,w\in A$ with $2w=u+v$ -- that is, an arithmetic progression. This is not quite a proof, because the arithmetic progression can be degenerate, but a slightly more careful version of the argument yields that the number of triforces exceeds the number of degenerate arithmetic progressions. Motivated by this result, it is natural to formulate the following general problem: how many edges can a 3-uniform hypergraph on $n$ vertices have if no $v$ vertices span $e$ or more edges? In particular, the well-known Brown-Erdős-Sós question asks whether if $e$ is fixed and $v=e+3$, then this number is $o(n^2)$. A well-known result in this direction, due to Sárközy and Selkow in 2004, is that the number of edges is $o(n^2)$ if $v=e+\lfloor \log_2 e\rfloor+2$. Note that when $e=3$, $\lfloor\log_2e\rfloor=1$, so this result generalizes the Ruzsa-Szemerédi $(6,3)$-theorem. This result has been improved more recently: in 2017 Solymosi and Solymosi obtained the conclusion if $e=10$ and $v=14$, instead of the 15 that would be given by the Sárközy-Selkow theorem, and Conlon, Gishboliner, Levanzov and Shapira obtained the first asymptotic improvement by showing that the conclusion holds if $v=e+\lceil 26\log e/\log\log e\rceil$. This paper concerns a related question: how large does $v$ have to be (as a function of $e$) for the bound to be not just $o(n^2)$ but $O(n^{2-c})$ for some positive constant $c$ (depending on $e$)? The main result is that it suffices if $v=\lfloor \log_2 e\rfloor +38$. Thus, their bound matches the Sárközy-Selkow bound (but not the Conlon-Gishboliner-Levanzov-Shapira bound) up to an additive constant. A bound of $e+2\log_2e+C$ was previously known, but quite a lot easier to prove and therefore not formally published anywhere -- it could perhaps be described as folklore -- but to remove the factor of 2 the authors use tools such as the sunflower lemma that had not previously made an appearance in this circle of ideas. One reason to be interested in power savings in the Brown-Erdős-Sós problem is an observation of Gowers and Long that a power saving for the $(9,5)$ problem would yield a positive solution to the following stubbornly open problem in additive combinatorics. **Problem.** _Do there exist constants $c, C>0$ such that for every $n$, every subset of $\{1,2,\dots,n\}$ of size at least $Cn^{1-c}$ contains distinct elements $x,y,z,w$ such that $2(x+y)=z+3w$?_ A straightforward modification of the proof of Roth's theorem yields a bound of $o(n^2)$ for this problem, but what makes the question interesting is that in the other direction it is _not_ possible to modify the Behrend construction to give a lower bound of the form $n^{1-o(1)}$.

In a recent breakthrough, Zhang proves that if G is an H-free graph with m edges, then G has a cut of size at least m/2+c_Hm^0.5001, making a significant step towards a well known conjecture of Alon, Bollobás, Krivelevich and Sudakov. We show that the methods of Zhang can be further boosted, and prove the following strengthening. If G is a graph with m edges and no clique of size m^1/2-δ, then G has a cut of size at least m/2+m^1/2+ε for some ε=ε(δ)>0. In addition, we sharpen another result of Zhang by proving that if G is an n-vertex m-edge graph with MaxCut of size at most m/2+n^1+ε (or its smallest eigenvalue λ_n satisfies |λ_n|≤ n^ε), then G is n^-ε-close to the disjoint union of cliques for some absolute constant ε>0.

The canonical Ramsey theorem of Erdős and Rado implies that for any graph H, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph K_N contains a monochromatic, lexicographic, or rainbow copy of H. The least such N is called the Erdős-Rado number of H, denoted by ER(H). Erdős-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős-Rado numbers of sparse graphs. For example, we prove that if H has bounded degree, then ER(H) is polynomial in |V(H)| if H is bipartite, but exponential in general. We also study the closely-related problem of constrained Ramsey numbers. For a given tree S and given path P_t, we study the minimum N such that every edge-coloring of K_N contains a monochromatic copy of S or a rainbow copy of P_t. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type.

We show that there exists an absolute constant γ>0 such that for every A⊆ℤ_>0 we have min_x∈ [0, 2π]∑_a∈ Acos(ax)≤ -Ω(|A|^γ). This gives the first polynomial bound for Chowla's cosine problem from 1965. To show this, we prove structural statements about graphs whose smallest eigenvalue is small in absolute value. As another application, we show that any graph G with m edges and no clique of size m^1/2-δ has a cut of size least m/2+m^1/2+ε for some ε=ε(δ)>0. This proves a weak version of a celebrated conjecture of Alon, Bollobás, Krivelevich, and Sudakov. Our proofs are based on novel spectral and linear algebraic techniques, involving subspace compressions and Hadamard products of matrices.

For a field F and integers d and k, a set A subset of F-d is called k-nearly orthogonal if its members are non-self-orthogonal and every k+1 vectors of A include an orthogonal pair. We prove that for every prime p there exists some delta=delta(p)>0, such that for every field F of characteristic p and for all integers k >= 2 and d >= k, there exists a k-nearly orthogonal set of at least d(delta & sdot;k)/(logk) vectors of F-d. The size of the set is optimal up to the log k term in the exponent. We further prove two extensions of this result. In the first, we provide a large set A of non-self-orthogonal vectors of F-d such that for every two subsets of A of size k+1 each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every k+1 vectors of the produced set A include & ell;+1 pairwise orthogonal vectors for an arbitrary fixed integer 1 <= & ell; <= k. The proofs involve probabilistic and spectral arguments and the hypergraph container method. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper we study the number of incidences between m points and n varieties in 𝔽^d, where 𝔽 is an arbitrary field, assuming the incidence graph contains no copy of K_s,s. We also consider the analogous problem for algebraically defined graphs and unit distance graphs. First, we prove that if 𝒫 is a set of m points and 𝒱 is a set of n varieties in 𝔽^D, each of dimension d and degree at most Δ, and in addition the incidence graph is K_s,s-free, then the number of incidences satisfies I(𝒫, 𝒱)≤ O_d,Δ, s(m^d/d+1 n+m). This bound is tight when s,Δ are sufficiently large with respect to d, with an appropriate choice of 𝔽=𝔽(m,n). We give two proofs of this upper bound, one based on the framework of the induced Turán problems and the other based on VC-dimension theory. In the second proof, we extend the celebrated result of Rónyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest. We also resolve the problem of finding the maximum number of unit distances which can be spanned by a set of n points 𝒫 in 𝔽^𝕕 whose unit-distance graph is K_s, s-free, showing that it is Θ_d,s(n^2-1/⌈ d/2⌉ +1). Finally, we obtain tight bounds on the maximum number of edges of a K_s, s-free algebraic graph defined over a finite field, thus resolving the Zarankiewicz problem for this class of graphs.

In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the threshold probability at which such graphs attain maximal connectivity. We slightly improve the lower bounds on the threshold probabilities, thus disproving two conjectures of Alon, Defant and Kravitz. We also improve the upper bound on the threshold probability in the case of random bipartite graphs, and obtain a tight bound up to a factor of $n^{o(1)}$. Further, we introduce a generalization of the notion of friends-and-strangers graphs in which vertices of the starting graphs are allowed to have multiplicities and obtain generalizations of previous results of Wilson and of Defant and Kravitz in this new setting.

We study and solve several problems in two closely related settings: set families in 2^[n] with many disjoint pairs of sets and low rank matrices with many zero entries. - More than 40 years ago, Daykin and Erdős asked for the maximum number of disjoint pairs of sets in a family F⊆ 2^[n] of size 2^(1/2+δ)n and conjectured it contains at most o(|F|^2) such pairs. This was proven by Alon and Frankl in 1985. In this paper we completely resolve this problem, proving an optimal dependence of the number of disjoint pairs on the size of family F. We also prove the natural variant of the Daykin-Erdős conjecture in which disjoint pairs are replaced by pairs with intersection λ≠ 0. - Motivated by a conjecture of Lovett related to the famous log-rank conjecture, Singer and Sudan asked to show that for two families A, B ⊆ 2^[n] with a positive constant fraction of set pairs (a,b)∈ A× B being disjoint, there are R⊂ A and S⊂ B such that all set pairs (r, s)∈ R× S are disjoint, and |R|≥ 2^-O(√(n))|A| and |S|≥ 2^-O(√(n))|B|. We prove this conjecture in a strong quantitative form. - We prove the following generalizations of the best known bounds for the log-rank conjecture. If M is an n× n non-negative integer matrix of rank r in which the average of the entries is ε≤ 1/2, then M contains an all-zero submatrix of size at least 2^-O(√(ε r))n. Unlike the known bounds for the log-rank conjecture, this result is optimal. Moreover, using similar methods, we also prove that any n× n matrix of rank r with entries from {0,…,t} contains a constant submatrix of size at least 2^-O(t√(r))n. Our proofs use probabilistic, entropy and discrepancy methods and explore connections to additive combinatorics and coding theory.

The study of counting point-hyperplane incidences in the d-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no s points are contained in the intersection of s hyperplanes, the currently best known upper bound on the number of incidences of m points and n hyperplanes in ℝ^d is O_d, s((mn)^1-1/(d+1)+m+n). This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers. In this paper, we propose a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in ℝ^d for many interesting values of m, n, d, e.g. when m=n and d is odd. Moreover, in finite fields our bounds are sharp as a function of m and n in every dimension. We also study the size of the largest complete bipartite graph in point-hyperplane incidence graphs with a given number of edges and obtain optimal bounds as well. Additionally, we study point-variety incidences and unit-distance problem in finite fields, and give tight bounds for both problems under a similar non-degeneracy assumption. We also resolve Zarankiewicz type problems for algebraic graphs. Our proofs use tools such as induced Turán problems, VC-dimension theory, evasive sets and Hilbert polynomials. Also, we extend the celebrated result of Rónyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest.

We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.