Consider the (almost surely) unique Radon partition of a set of n random Gaussian vectors in ℝ^n-2; choose one of the two parts of this partition uniformly at random, and for 0 ≤ k ≤ n, let p_k denote the probability that it has size k. In this paper, we prove strong unimodality results for the distribution (p_0,…,p_n).

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. •Panchromatic Graphs: For fixed $k\in \mathbb{N}$ , a $k$ -panchromatic graph is, roughly speaking, a balanced bipartite graph with one partition class equipartitioned into $k$ colour classes in which the common neighbourhoods of panchromatic $k$ -sets of vertices are much larger than those of $k$ -sets that repeat a colour. The question of their existence was raised by Karthik and Manurangsi [Combinatorica 2020]. •Threshold Graphs: For fixed $k\in \mathbb{N}$ , a $k$ -threshold graph is, roughly speaking, a balanced bipartite graph in which the common neighbourhoods of $k$ -sets of vertices on one side are much larger than those of ( $k+1$ )-sets. The question of their existence was raised by Lin [JACM 2018]. Concretely, we provide probability distributions over graphs from which we can efficiently sample these objects in near linear time. These probability distributions are defined via varieties cut out by (carefully chosen) random polynomials, and the analysis of these constructions relies on machinery from algebraic geometry (such as the Lang-Weil estimate, for example). The technical tools developed to accomplish this might be of independent interest. As applications of our constructions, we show the following conditional time lower bounds on the parameterized set intersection problem where, given a collection of $n$ sets over universe [ $n$ ] and a parameter $k$ , the goal is to find $k$ sets with the largest intersection. •Assuming ETH, for any computable function $F:\mathbb{N}\rightarrow \mathbb{N}$ , no $n^{o(k)}$ -time algorithm can approximate the parameterized set intersection problem up to factor $F(k)$ . This improves considerably on the previously best-known result under ETH due to Lin [JACM 2018], who ruled out any $n^{o(\sqrt{k})}$ time approximation algorithm for this problem. •Assuming SETH, for every $\varepsilon > 0$ and any computable function $F:\mathbb{N} \rightarrow \mathbb{N}$ , no $n^{k-\varepsilon}$ -time algorithm can approximate the parameterized set intersection problem up to factor $F(k)$ . No result of comparable strength was previously known under SETH, even for solving this problem exactly. Both these time lower bounds are obtained by composing panchromatic graphs with instances of the coloured variant of the parameterized set intersection problem (for which tight lower bounds were previously known).

We study several basic problems about colouring the p-random subgraph G_p of an arbitrary graph G, focusing primarily on the chromatic number and colouring number of G_p. In particular, we show that there exist infinitely many k-regular graphs G for which the colouring number (i.e., degeneracy) of G_1/2 is at most k/3 + o(k) with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least k/2 - o(k).

We study how many copies of a graph F that another graph G with a given number of cliques is guaranteed to have. For example, one of our main results states that for all t \geq 2, if G is an n-vertex graph with kn3/2 triangles and k is sufficiently large in terms of t, then G contains at 2t2 5t-2 least \Omega(min\{ktn3/2, k 3t-1 n 3t-1 \} ) copies of K2,t, and, furthermore, we show that these bounds are essentially best possible provided that either k \geq n1/2t or certain bipartite analogues of well-known conjectures for Tura'\n numbers hold.

Given a random binary picture $P_n$ of size $n$, i.e., an $n\times n$ grid filled with zeros and ones uniformly at random, when is it possible to reconstruct $P_n$ from its $k$-deck, i.e., the multiset of all its $k\times k$ subgrids? We demonstrate ``two-point concentration'' for the reconstruction threshold by showing that there is an integer $k_c(n) \sim (2 \log n)^{1/2}$ such that if $k > k_c$, then $P_n$ is reconstructible from its $k$-deck with high probability, and if $k < k_c$, then with high probability, it is impossible to reconstruct $P_n$ from its $k$-deck. The proof of this result uses a combination of interface-exploration arguments and entropic arguments.

Let ϵ_1,…,ϵ_n be a sequence of independent Rademacher random variables. We prove that there is a constant c>0 such that for any unit vectors v_1,…,v_n∈ℝ^2, [||ϵ_1 v_1+…+ϵ_n v_n||_2 ≤√(2)]≥c/n. This resolves the only remaining conjecture from the seminal paper of Erdős on the Littlewood–Offord problem, and it is sharp both in the sense that the constant √(2) cannot be reduced and that the magnitude n^-1 is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions.

We show that the natural directed analogues of the KKL theorem [6] and the Eldan-Gross inequality [4] from the analysis of Boolean functions fail to hold. This is in contrast to several other isoperimetric inequalities on the Boolean hypercube (such as the Poincare inequality, Margulis's inequality [9] and Talagrand's inequality [14]) for which directed strengthenings have recently been established.

Let $\epsilon_{1},\ldots,\epsilon_{n}$ be a sequence of independent Rademacher random variables. We prove that there is a constant $c>0$ such that for any unit vectors $v_1,\ldots,v_n\in \mathbb{R}^2$, $$\Pr\left[||\epsilon_1 v_1+\ldots+\epsilon_n v_n||_2 \leq \sqrt{2}\right]\geq \frac{c}{n}.$$ This resolves the only remaining conjecture from the seminal paper of Erd\H{o}s on the Littlewood--Offord problem, and it is sharp both in the sense that the constant $\sqrt{2}$ cannot be reduced and that the magnitude $n^{-1}$ is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions.

A family of sets A$A$ is said to be an antichain if x⊄y$x\not\subset y$ for all distinct x,y∈A$x,y\in A$ , and it is said to be a distance‐ r$r$ code if every pair of distinct elements of A$A$ has Hamming distance at least r$r$ . Here, we prove that if A⊂2[n]$A\subset 2^{[n]}$ is both an antichain and a distance‐ r$r$ code, then |A|=Or(2nn−1/2−⌊(r−1)/2⌋)$|A| = O_r(2^n n^{-1/2 - \lfloor (r-1)/2\rfloor } )$ . This result, which is best‐possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood–Offord theory; for example, our result gives a short combinatorial proof of Hálasz's theorem, while all previously known proofs of this result are Fourier‐analytic.

We elucidate the relationship between the threshold and the expectation‐threshold of a down‐set. Qualitatively, our main result demonstrates that there exist down‐sets with polynomial gaps between their thresholds and expectation‐thresholds; in particular, the logarithmic gap predictions of Kahn–Kalai and Talagrand (recently proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about up‐sets do not apply to down‐sets. Quantitatively, we show that any collection 𝒢 of graphs on [n] that covers the family of all triangle‐free graphs on [n] satisfies the inequality ∑G∈𝒢exp(−δe(Gc)/n)<1/2 for some universal δ>0 , and this is essentially best‐possible.

Our first main result is the following basic fact about simplicial complexes: for each k is an element of N, there exists an exponent lambda(k) >= k(-2)(k2) such that for any k-complex S, every k-complex on n >= n(0) (S) vertices with at least n(k+1-lambda k) facets contains a homeomorphic copy of S. The existence of these exponents was suggested by Linial in 2006 but was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of lambda(1) is a result of Mader from 1967, and the existence of lambda(2) was established by Keevash-Long-Narayanan-Scott in 2020. We deduce this geometric theorem from a purely combinatorial result about trace-bounded hypergraphs, where an r-partite r-graph H with partition classes V-1, V-2, ..., V-r is said to be d-trace-bounded if for each 2 <= i <= r, all the vertices of V-i have degree at most d in the trace of H on V-1 boolean OR V-2 boolean OR ... boolean OR V-i. Our second main result is the following fact about degenerate trace-bounded hypergraphs: for all r >= 2 and d is an element of N, there exists an exponent alpha(r,d) >= (5rd)(1-r) such that for any d-trace-bounded r-partitc r-graph H, every r-graph on n >= n(0)(H) vertices with at least n(r-alpha r,d) edges contains a copy of H. This strengthens a theorem of Conlon-Fox-Sudakov from 2009 who showed that a similar result holds for r-partite r-graphs H satisfying the stronger hypothesis that the vertex-degrees in all but one of its partition classes are bounded (in H, as opposed to in its traces).

Simplicial homeomorphs and trace-bounded hypergraphs, Discrete Analysis 2022:6, 12 pp. A well-known result of Mader from 1967 states that for every finite graph $H$ there exists a positive integer $d$ such that every graph of average degree at least $d$ contains a subdivision of $H$ -- that is, a copy of $H$ where each edge may be replaced by a path (and the paths do not intersect). Of course, this follows from the special case where $H$ is a complete graph. The result was motivated by connections with a famous conjecture of Hadwiger, which states that if $G$ does not contain the complete graph $K_t$ as a minor, then the chromatic number of $G$ is less than $t$. Since no planar graph contains a $K_5$ minor, this is a generalization of the four-colour theorem. This paper concerns generalizations of Mader's theorem to $k$-dimensional simplicial complexes. It has been known for some time that for any such complex $K$ there exist $\lambda>0$ and $C$ such that every $k$-dimensional simplicial complex on $n$ vertices with at least $Cn^{k+1-\lambda})$ faces of dimension $k$ contains a complex homeomorphic to $K$. Note that each face of dimension $k$ is determined by $k+1$ vertices, so the trivial upper bound for the number of faces is $\binom n{k+1}$, and therefore this result is saying that the trivial upper bound can be beaten by a power of $n$. Given a $k$-dimensional complex $K$, let us define $\lambda_K$ to be the supremum over all $\lambda$ such that every $k$-dimensional complex that contains no homeomorphic copy of $K$ has $O(n^{k+1-\lambda})$ faces of dimension $k$. When $k=1$, Mader's theorem tells us that $\lambda_K=1$ for every $K$. (In this case, $K$ is a graph.) In particular, $\lambda_K$ is independent of $K$ that one is trying to find. It is therefore natural to ask whether the same is true in higher dimensions. That is, is it true that for each $k$ there exists $\lambda_k>0$ such that $\lambda_K\geq\lambda_k$ for every $k$-dimensional complex $K$? Before this paper it was shown by Peter Keevash, Alexander Scott and the first two authors of this paper that $\lambda_2$ exists and is at least $1/5$. The conjectured best possible value for $\lambda_2$ is 1/2, but this is still an open problem. The main result of this paper is a proof that $\lambda_k\geq k^{-2k^2}$ for every $k$, so there is indeed a uniform bound for every fixed $k$. However, determining the correct dependence on $k$ (even asymptotically) remains a challenging open problem. **Question for authors: is it known that $\lambda_k\to 0$? I couldn't find that in the paper, though I haven't looked super-hard for it.** The proof of the result goes via a Turán-type statement. The basic idea is to take a complex $K$ and construct its "canonical subdivision". This is the combinatorial version of the natural geometric subdivision one obtains by taking the centres of all the faces of every dimension as the vertices and joining $k+1$ of them whenever they form the centres of the faces of a $k$-dimensional "flag" (that is, sequence of faces $F_0\subset F_1\subset\dots\subset F_k$ where $F_i$ has dimension $i$). One can check easily that the $k$-dimensional faces of a canonical subdivision $\tilde K$ of $K$ form a $(k+1)$-partite $(k+1)$-uniform hypergraph (the vertex sets corresponding to the dimensions of the faces of $K$ of which the vertices of $\tilde K$ are the centres). Furthermore, this hypergraph has the following property. If $i>0$ and we take a vertex $v_i$ from the $i$th vertex class $V_i$, then the number of ways of choosing a vertices $v_0,v_1,\dots,v_{i-1}$ from the vertex classes $V_0,V_1,\dots,V_{i-1}$ in such a way that $v_0,v_1,\dots,v_i$ is contained in one of the faces of the hypergraph is at most $(i+1)!$, since the $i$-dimensional face that contains $v_i$ has $i+1$ faces of dimension $i-1$. (For instance, a tetrahedron is 3-dimensional and has four 2-dimensional faces. Note that the authors denote the vertex sets by $V_1,\dots,V_{k+1}$ instead of $V_0,V_1,\dots,V_k$.) In particular, this number is at most $(k+1)!$. In the terminology of the paper, this means that the hypergraph is $(k+1)!$-_trace bounded_. It therefore suffices to prove the Turán-type result that every $(k+1)$-uniform hypergraph that does not contain a subhypergraph isomorphic to some fixed $d$-trace bounded hypergraph has $O(n^{k+1-\lambda_k})$ hyperedges. Note that since a trace-bounded hypergraph is $(k+1)$-partite, it is reasonable to expect a power improvement on the trivial bound of $\binom n{k+1}$, just as in the graph case a graph that does not contain a fixed bipartite graph has $O(n^{2-\epsilon})$ edges for some positive $\epsilon$ (though with the difference that we are talking about an arbitrary member of a class of hypergraphs rather than a single hypergraph and we want a power that depends just on $d$). The authors show that for every $r$ and $d$ there exists $\alpha>0$ such that if $H$ is a $d$-trace bounded $r$-partite $r$-uniform hypergraph, then for sufficiently large $n$, every $r$-uniform hypergraph that does not contain $H$ has at most $n^{r-\alpha}$ hyperedges. The bound they obtain for $\alpha$ is $(5rd)^{-(r-1)}$. This result generalizes an earlier result of Conlon, Fox and Sudakov, which uses a stronger hypothesis on $H$ than trace boundedness (and obtains a slightly better bound). For the purposes of this paper, it is essential that trace boundedness is sufficient. Our first main result is the following basic fact about simplicial complexes: for each $k \in \N$, there exists an exponent $\lambda_k \ge k^{-2k^2}$ such that for any $k$-complex $\SS$, every $k$-complex on $n \ge n_0(\SS)$ vertices with at least $n^{k+1 - \lambda_k}$ facets contains a homeomorphic copy of $\SS$. The existence of these exponents was suggested by Linial in 2006 but was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $\lambda_1$ is a result of Mader from 1967, and the existence of $\lambda_2$ was established by Keevash--Long--Narayanan--Scott in 2020. We deduce this geometric theorem from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, \dots, V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 \cup V_2 \cup \dots \cup V_i$. Our second main result is the following fact about degenerate trace-bounded hypergraphs: for all $r \ge 2$ and $d\in\N$, there exists an exponent $\alpha_{r,d} \ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n \ge n_0(H)$ vertices with at least $n^{r - \alpha_{r,d}}$ edges contains a copy of $H$. This strengthens a theorem of Conlon--Fox--Sudakov from 2009 who showed that a similar result holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).

Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic number at least $k$. This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon-Kleitman-Thomassen-Saks-Seymour from 1987 showing that $f(k) = O(k^3)$, and of Chudnovsky-Penev-Scott-Trotignon from 2013 showing that $f(k) = O(k^2)$. Our methods are robust enough to handle list colouring as well: we also show that for each $k \in \mathbb{N}$, there exists an $f_\ell(k) \le 4k$ such that every graph $G$ with list chromatic number at least $f_\ell(k)+1$ contains a subgraph $H$ with both connectivity and list chromatic number at least $k$. This result is again best-possible up to multiplicative constants; here, unlike with $f(\cdot)$, even the existence of $f_\ell(\cdot)$ appears to have been previously unknown.

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface $\mathscr{S}$, we show that any two-dimensional simplicial complex on $n$ vertices in which each pair of vertices belongs to at least $n/3 + o(n)$ facets contains a homeomorph of $\mathscr{S}$ spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on $n$ vertices with minimum codegree exceeding $n/3+o(n)$ contains a spanning triangulation of the $2$-sphere.

We show that the natural directed analogues of the KKL theorem [KKL88] and the Eldan--Gross inequality [EG20] from the analysis of Boolean functions fail to hold. This is in contrast to several other isoperimetric inequalities on the Boolean hypercube (such as the Poincare inequality, Margulis's inequality [Mar74] and Talagrand's inequality [Tal93]) for which directed strengthenings have recently been established.

AbstractWe prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight (for complete graphs), and improves earlier estimates of Alon from 1990 and Kostochka from 1995 by a factor of about $1/n$ (for dense graphs). We additionally show that an analogous bound holds for the weighted spanning tree enumerator of a (nonnegatively) weighted graph as well.

Proving a conjecture of Talagrand, a fractional version of the "expectation-threshold" conjecture of Kalai and the second author, we show that p(c) (F) = O(q(f) (T) log l(F)) for any increasing family F on a finite set X, where p(c)(F) and q(f)(F) are the threshold and "fractional expectation-threshold" of F, and l(F) is the maximum size of a minimal member of F. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson Kahn Vu), bounded degree spanning trees (Mont-gomery), and bounded degree graphs (new). We also resolve (and vastly extend) the "axial" version of the random multi -dimensional assignment problem (earlier considered by Martin-Mezard-Rivoire and Frieze-Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdos-Rado "Sunflower Conjecture."

We present a principled technique for reducing the lattice and matrix size in some applications of Coppersmith's lattice method for finding roots of modular polynomial equations. It relies on extrapolating patterns from the actual behavior of Coppersmith's attack for smaller parameter sizes, which can be thought of as “focus group” testing. When applied to the small-exponent RSA problem, our technique reduces lattice dimensions and consequently running times, and hence can be applied to a wider range of exponents. Moreover, in many difficult examples our attack is not only faster but also more successful in recovering the RSA secret key. We include a discussion of subtleties concerning whether or not existing metrics (such as enabling condition bounds) are decisive in predicting the true efficacy of attacks based on Coppersmith's method. Finally, indications are given which suggest certain lattice basis reduction algorithms (such as Nguyen-Stehlé's L2) may be particularly well-suited for Coppersmith's method.

A family of vectors in [ k ] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [ k ] n invariant under a transitive group of symmetries is o ( k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.