Abstract. Schur’s theorem states that in any [Formula: see text]-coloring of the set of integers [Formula: see text] there is a monochromatic solution to [Formula: see text], provided [Formula: see text] is sufficiently large. Abbott and Wang studied the size of the largest subset of [Formula: see text] such that there is a [Formula: see text]-coloring avoiding a monochromatic [Formula: see text]. This led to the exploration of related problems, such as the minimum number of monochromatic [Formula: see text] in [Formula: see text]-colorings of [Formula: see text] and the probability threshold for a random subset of [Formula: see text] to have a monochromatic [Formula: see text] in any [Formula: see text]-coloring. In this paper, we study natural generalizations of these problems to products [Formula: see text], in deterministic, random, and randomly perturbed environments.

We present a fully computer-assisted proof system for solving a particular family of problems in Extremal Combinatorics. Existing techniques using Flag Algebras have proven powerful in the past, but have so far lacked a computational counterpart to derive matching constructive bounds. We demonstrate that common search heuristics are capable of finding constructions far beyond the reach of human intuition. Additionally, the most obvious downside of such heuristics, namely a missing guarantee of global optimality, can often be fully eliminated in this case through lower bounds and stability results coming from the Flag Algebra approach. To illustrate the potential of this approach, we study two related and well-known problems in Extremal Graph Theory that go back to questions of Erdős from the 60s. Most notably, we present the first major improvement in the upper bound of the Ramsey multiplicity of K_4 in 25 years, precisely determine the first off-diagonal Ramsey multiplicity number, and settle the minimum number of independent sets of size four in graphs with clique number strictly less than five.

For graphs $G$ and $H$, let $G \overset{\mathrm{rb}}{\longrightarrow} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, Škorić and Steger [J. Combin. Theory Ser. B 124 (2017),1-38], we determine the threshold for $G(n,p) \overset{\mathrm{rb}}{\longrightarrow} C_\ell$ for cycles $C_\ell$ of any given length $\ell \geq 4$.

Recent advances in noiseless non-adaptive group testing have led to a precise asymptotic characterization of the number of tests required for high-probability recovery in the sublinear regime$k = n^{\theta }$(with$\theta \in (0,1)$), with$n$individuals among which$k$are infected. However, the required number of tests may increase substantially under real-world practical constraints, notably including bounds on the maximum number$\Delta $of tests an individual can be placed in, or the maximum number$\Gamma $of individuals in a given test. While previous works have given recovery guarantees for these settings, significant gaps remain between the achievability and converse bounds. In this paper, we substantially or completely close several of the most prominent gaps. In the case of$\Delta $-divisible items, we show that the definite defectives (DD) algorithm coupled with a random regular design is asymptotically optimal in dense scaling regimes, and optimal to within a factor of e more generally; we establish this by strengthening both the best known achievability and converse bounds. In the case of$\Gamma $-sized tests, we provide a comprehensive analysis of the regime$\Gamma = \Theta (1)$, and again establish a precise threshold proving the asymptotic optimality of SCOMP (a slight refinement of DD) equipped with a tailored pooling scheme. Finally, for each of these two settings, we provide near-optimal adaptive algorithms based on sequential splitting, and provably demonstrate gaps between the performance of optimal adaptive and non-adaptive algorithms.

We study the problem of finding pairwise vertex-disjoint copies of the ℓ-vertex cycle Cℓ in the randomly perturbed graph model, which is the union of a deterministic n-vertex graph G and the binomial random graph G(n, p). For ℓ ≥ 3 we prove that asymptotically almost surely G U G(n, p) contains min{δ(G), min{δ(G), [n/l]} pairwise vertex-disjoint cycles Cℓ, provided p ≥ C log n/n for C sufficiently large. Moreover, when δ(G) ≥ αn with 0 ≤ α/l and G and is not ‘close’ to the complete bipartite graph Kαn,(1 - α)n, then p ≥ C/n suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that p ≥ C/n suffices when α > n/l for finding [n/l] cycles Cℓ. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson-Kahn-Vu Theorem for Cℓ-factors and the resolution of the El-Zahar Conjecture for Cℓ-factors by Abbasi.

Maker-Breaker games are played on a hypergraph $(X,\mathcal{F})$, where $\mathcal{F} \subseteq 2^X$ denotes the family of winning sets. Both players alternately claim a predefined amount of edges (called bias) from the board $X$, and Maker wins the game if she is able to occupy any winning set $F \in \mathcal{F}$. These games are well studied when played on the complete graph $K_n$ or on a random graph $G_{n,p}$. In this paper we consider Maker-Breaker games played on randomly perturbed graphs instead. These graphs consist of the union of a deterministic graph $G_\alpha$ with minimum degree at least $\alpha n$ and a binomial random graph $G_{n,p}$. Depending on $\alpha$ and Breaker's bias $b$ we determine the order of the threshold probability for winning the Hamiltonicity game and the $k$-connectivity game on $G_{\alpha}\cup G_{n,p}$, and we discuss the $H$-game when $b=1$.