In this paper we show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multi-type branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the so-called random d-process, resolving a problem of Warnke and Wormald for large d.

The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for d-regular degree sequences D_n, the final graph of this process is similar to a uniform random d-regular graph. In this paper we show that, for degree sequences D_n that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence D_n. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before).

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph G_{n,p} for most edge-probabilities p in the range n^{-2/5} \ll p \ll 1. This resolves open problems and questions of Lichev, Mitsche and Warnke as well as Alon and Krievelevich. One major proof difficulty stems from high-degree vertices, which prevent maximal cliques in their neighborhoods: we deal with these vertices by an intricate union bound argument, that combines the probabilistic method with new degree counting arguments in order to enable Janson's inequality. This way we determine the asymptotics of the clique chromatic number of G_{n,p} in some ranges, and discover a surprising new phenomenon that contradicts earlier predictions for edge-probabilities p close to n^{-2/5}.

We study the algorithmic problem of finding a large independent set in the Erdös-Rényi random graph G(n,p). For constant p and b=1/(1-p), the largest independent set has size 2log_b n, while a simple greedy algorithm - revealing vertices sequentially and making decisions based only on previously seen vertices - finds an independent set of size log_b n. In his seminal 1976 paper, Karp challenged to either improve this guarantee or establish its hardness. Decades later, this problem remains open - one of the most prominent algorithmic problems in the theory of random graphs. In this paper, we establish that a broad class of online algorithms fails to find an independent set of size (1+ε)log_b n whp. This class includes Karp's algorithm as a special case, and extends it by allowing the algorithm to query exceptional edges, not yet "seen" by the algorithm. Our lower bound holds for p∈ [d/n,1-n^-1/d]. In the dense regime (constant p), we also prove that our result is asymptotically tight with respect to the number of exceptional edges queried, by designing an online algorithm which beats the half-optimality threshold when the number of exceptional edges slightly exceeds our bound. Our result provides evidence for the algorithmic hardness of Karp's problem, by supporting the conjectured optimality of the greedy algorithm and establishing it within the class of online algorithms. Our proof relies on a refined analysis of the geometric structure of large independent sets, establishing a variant of the Overlap Gap Property (OGP). While OGP has predominantly served as a barrier to stable algorithms, online algorithms are inherently unstable, necessitating new ideas. Our proof refines the OGP framework by incorporating several new ideas (including temporal interpolation paths and stopping-times) that we expect to be useful for other online models.

We consider two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities p_1,p_2. In particular, (i) we prove a sharp threshold result for the appearance of G_n,p_1 as an induced subgraph of G_N,p_2, (ii) we show two-point concentration of the size of the maximum common induced subgraph of G_N, p_1 and G_N,p_2, and (iii) we show that the number of induced copies of G_n,p_1 in G_N,p_2 has an unusual limiting distribution. These results confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble, and resolve several open problems of Chatterjee and Diaconis. The proofs are based on careful refinements of the first and second moment method, using extra twists to (a) take some non-standard behaviors into account, and (b) work around the large variance issues that prevent standard applications of these methods.

This work studies the typical structure of sparse H-free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph H. Extending the seminal result of Osthus, Pr & ouml;mel, and Taraz that addressed the case where H is an odd cycle, Balogh, Morris, Samotij, and Warnke proved that, for every r >= 3, the structure of a random Kr+1-free graph with n vertices and m edges undergoes a phase transition when m crosses an explicit (sharp) threshold function m(r)(n). They conjectured that a similar threshold phenomenon occurs when Kr+1 is replaced by any strictly 2-balanced, edge-critical graph H. In this paper, we resolve this conjecture. In fact, we prove that the structure of a typical H-free graph undergoes an analogous phase transition for every H in a family of vertex-critical graphs that includes all edge-critical graphs.

We show that the domination number of the binomial random graph G_{n,p} with edge-probability p is concentrated on two values for p \ge n^{-2/3+\eps}, and not concentrated on two values for general p \le n^{-2/3}. This refutes a conjecture of Glebov, Liebenau and Szabo, who showed two-point concentration for p \ge n^{-1/2+\eps}, and conjectured that two-point concentration fails for p \ll n^{-1/2}. The proof of our main result requires a Poisson type approximation for the probability that a random bipartite graph has no isolated vertices, in a regime where standard tools are unavailable (as the expected number of isolated vertices is relatively large). We achieve this approximation by adapting the proof of Janson's inequality to this situation, and this adaptation may be of broader interest.

Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.

Gunby-He-Narayanan showed that the logarithmic gap predictions of Kahn-Kalai and Talagrand (proved by Park-Pham and Frankston-Kahn-Narayanan-Park) about thresholds of up-sets do not apply to down-sets. In particular, for the down-set of triangle-free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down-sets of F-free graphs.

The Prague dimension of graphs was introduced by Nešetřil, Pultr and Rödl in the 1970s. Proving a conjecture of Füredi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/log n for constant edge-probabilities. The main new proof ingredient is a Pippenger–Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n) .

We consider maximum rooted tree extension counts in random graphs, i.e., we consider M_n = \max_v X_v where X_v counts the number of copies of a given tree in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the random graph is not too sparse, specifically when the edge probability p=p(n) satisfies p(1-p)n \gg \log n. The problem is more difficult in the sparser regime 1 \ll pn \ll \log n, where we determine the asymptotics of M_n for specific classes of trees. Interestingly, here our large deviation type optimization arguments reveal that the behavior of M_n changes as we vary p=p(n), due to different mechanisms that can make the maximum large.

We present a simple inductive proof of the Lagrange Inversion Formula.

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around p≈n−1/2$$ p\approx {n}^{-1/2} $$ the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ changes by nΩ(1)$$ {n}^{\Omega (1)} $$ when we increase the edge‐probability p$$ p $$ by no(1)$$ {n}^{o(1)} $$ , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph Gn,p$$ {G}_{n,p} $$ around edge‐probability p≈n−1/2$$ p\approx {n}^{-1/2} $$ . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of Gn,p$$ {G}_{n,p} $$ up to logarithmic factors for any edge‐probability p$$ p $$ .

We present a refinement of the classical alteration method for constructing H $H$-free graphs for suitable edge-probabilities p $p$, we show that removing all edges in H $H$-copies of the binomial random graph Gn,p ${G}_{n,p}$ does not significantly change the independence number. This differs from earlier alteration approaches of Erdos and Krivelevich, who obtained similar guarantees by removing one edge from each H $H$-copy (instead of all of them). We demonstrate the usefulness of our refined alternation method via two applications to online graph Ramsey games, where it enables easier analysis.

Costas arrays are useful in radar and sonar engineering, and many other settings in which optimal 2-D autocorrelation is needed: they are permutation matrices in which the vectors joining different pairs of ones are all distinct. We prove that the density of Costas arrays among permutation matrices decays exponentially, solving a core problem in the theory of Costas arrays. The proof combines ideas from random graph theory with tools from probabilistic combinatorics.

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.

We study the phase transition of the random d -process, which is a time-evolving random graph model with bounded degrees: starting with an empty graph on n vertices, new random edges are added step-by-step so that the maximum degree remains at most d . We show that the d -process undergoes a giant component phase transition for fixed d (cid:62) 3, solving a problem of Wormald from 1999. We also show that the giant component has a non-trivial distribution at the end of the 2-process, verifying a conjecture of Bali´nska and Quintas from 1993. Our techniques show that the critical point of the phase transition in fact equals the blow-up point of the susceptibility, which in turn is determined by O ( d 2 ) many differential equations. Our results extend to many hypergraph generalizations of the d -process.

In this paper we solve two problems of Esperet, Kang and Thomassé as well as Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii) van der Waerden numbers. Each time random greedy algorithms allow us to go beyond the Lovász Local Lemma or alteration method used in previous work, illustrating the power of the algorithmic approach to the probabilistic method.

We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the configuration model.

A sunflower with p petals consists of p sets whose pairwise intersections are identical. The goal of the sunflower problem is to find the smallest r=r(p,k) such that any family of rk distinct k-element sets contains a sunflower with p petals. Building upon a breakthrough of Alweiss, Lovett, Wu and Zhang from 2019, Rao proved that r=O(plog(pk)) suffices; this bound was reproved by Tao in 2020. In this short note we record that r=O(plogk) suffices, by using a minor variant of the probabilistic part of these recent proofs.