The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple $k$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$ -partite, while, for $c$ bounded away from $0$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$ .

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$ -approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

Perturbations to the gut microbiome are implicated in altered neurodevelopmental trajectories that may shape life span risk for emotion dysregulation and affective disorders. However, the sensitive periods during which the microbiome may influence neurodevelopment remain understudied. We investigated relationships between gut microbiome composition across infancy and temperament at 12 months of age. In 67 infants, we examined if gut microbiome composition assessed at 1–3 weeks, 2, 6, and 12 months of age was associated with temperament at age 12 months. Stool samples were sequenced using the 16S Illumina MiSeq platform. Temperament was assessed using the Infant Behavior Questionnaire-Revised (IBQ-R). Beta diversity at age 1–3 weeks was associated with surgency/extraversion at age 12 months. Bifidobacterium and Lachnospiraceae abundance at 1–3 weeks of age was positively associated with surgency/extraversion at age 12 months. Klebsiella abundance at 1–3 weeks was negatively associated with surgency/extraversion at 12 months. Concurrent composition was associated with negative affectivity at 12 months, including a positive association with Ruminococcus-1 and a negative association with Lactobacillus. Our findings support a relationship between gut microbiome composition and infant temperament. While exploratory due to the small sample size, these results point to early and late infancy as sensitive periods during which the gut microbiome may exert effects on neurodevelopment.

In this short note we prove that every tournament contains the k-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when k=2, showing that there is always a square of a directed path of length , which is best possible.

The hedgehog Ht is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).

Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).

On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

We provide a deterministic algorithm that finds, in ɛ-O(1)n2 time, an ɛ-regular Frieze–Kannan partition of a graph on n vertices. The algorithm outputs an approximation of a given graph as a weighted sum of ɛ-O(1) many complete bipartite graphs.

As a corollary, we give a deterministic algorithm for estimating the number of copies of H in an n-vertex graph G up to an additive error of at most ɛnv(H), in time ɛ-OH(1)n2.

OBJECTIVES/SPECIFIC AIMS: Therapeutic hypothermia (TH) is a neuroprotective therapy regularly used in newborn infants following traumatic births. The infant’s temperature is maintained at 33.5°C for 72 hours by a cooling blanket upon which the infant is placed. Parents are not permitted to hold their infant while TH is ongoing due to concerns for unintentional rewarming or accidental dislodging of catheters or other monitoring equipment. Our prior qualitative research with nurse and parent interviews described the inability to hold an infant during TH as a significant source of stress. We assessed the feasibility of a 30-minute period of maternal holding for infants being actively treated with TH and assessed both the maternal experience of holding and the nurse experience of supporting holding. METHODS/STUDY POPULATION: This was a feasibility study employing a mixed-methods approach. Inclusion criteria were gestational age at birth of 35 weeks or greater, absence of clinical or electrographic seizures during the first 24 hours of TH, and designation as “clinically stable” by the attending neonatologist with the infant on room air, nasal cannula, or continuous positive airway pressure. Quantitative data were obtained from vital sign monitoring every 2 minutes before, during and after holding and from maternal and nurse research surveys. Qualitative data were obtained from nurse surveys. Infant rewarming was prevented through use of a thin foam insulating barrier placed between mother and infant during holding. Adverse events were defined as a change in infant temperature greater than 0.5°C above or below 33.5°C, accidental dislodging of central lines/disruption of EEG leads or early termination of holding due vital sign instability present for greater than 2 recorded measurements including infant bradycardia defined as heart rate less than 80 beats per minute, hypotension defined as mean arterial pressure less than 40 mmHg or oxygen saturation of less than 93%. RESULTS/ANTICIPATED RESULTS: There were 10 newborn infants undergoing TH for neonatal encephalopathy (median gestational age 39.4 weeks) and their mothers (median age=31 years) were recruited. Infants remained on the hypothermia blanket during holding and were transferred safely to their mother’s arms without medical equipment malfunction/dislodgement. Holding occurred at a median of 47 hours of life. The mean temperature prior to holding was 33.4°C and at completion of holding the mean temperature was 33.5°C (p=0.18). There were no significant bradycardia, hypotension or oxygen desaturation events. In total, 80% of mothers reported difficulty bonding with their baby prior to holding and 90% reported a high level of stress before holding. After holding, all mothers felt their bond was “stronger” or “much stronger” and all felt “less stressed” or “much less stressed.” After holding, 75% of nurses reported that they felt a more positive emotional response to the infant. One nurse stated, “being a part of this emotional experience made me feel closer and more connected to this family and gave me a different perspective on just what they had been dealing with and feeling since giving birth to their child.” In free text responses, on 5 separate occasions, nurses commented on the relaxed, calmed or less irritable appearance of the infant while being held during TH. DISCUSSION/SIGNIFICANCE OF IMPACT: In this sample of term infants treated with TH, a 30-minute period of maternal holding was not associated with increased temperature or other adverse events. Holding during TH was associated with extremely positive feedback from mothers and nurses. Future larger studies could consider assessing the impact of holding on endocrinological markers of stress and bonding, on infant glycemic control, on breastfeeding success rates, and the impact of earlier and improved bonding on the developmental outcomes of children held during their treatment with TH. Increasing the duration of holding and allowing both parents to hold on more than one occasion during the 72 hours of TH may increase the proposed benefits of this intervention.

The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are ε-far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with ‘constant’ query complexity, depending only on ε and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques is often enormous and impractical. It remains a major open problem if better bounds hold.

Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ε query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

This paper describes select results of a longitudinal study of 62 mild to moderate Alzheimer's disease (AD) patients, in comparison to 60 age-matched healthy controls. Initial neurologic, radiologic, psychiatric, laboratory and cognitive examinations, required two full days, followed by one-day examinations at annual intervals. Of the total original sample, 31 AD patients and 39 controls could actually be followed for three annual examinations. Cognitive examination data confirmed cross-sectional (group discriminative) validity of memory and language measures, and showed the expected longitudinal deterioration in the AD sample, with controls maintaining consistent performance over the three years. However, those measures showing largest group differences at initial examination were not the best for tracking patient deterioration over time. Implications of these results for the selection of cognitive assessment measures are discussed.

A graph on n vertices is ε-far from a property $\mathcal{P}$ if one has to add or delete from it at least εn2 edges to get a graph satisfying $\mathcal{P}$. A graph property $\mathcal{P}$ is strongly testable if for every fixed ε > 0 it is possible to distinguish, with one-sided error, between graphs satisfying $\mathcal{P}$ and ones that are ε-far from $\mathcal{P}$ by inspecting the induced subgraph on a random subset of at most f(ε) vertices. A property is easily testable if it is strongly testable and the function f is polynomial in 1/ε, otherwise it is hard. We consider the problem of characterizing the easily testable graph properties, which is wide open, and obtain several results in its study. One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard. On the other hand, we show that being a cograph, or equivalently, induced P3-freeness where P3 is a path with 3 edges, is easily testable. This settles one of the two exceptional graphs, the other being C4 (and its complement), left open in the characterization by the first author and Shapira of graphs H for which induced H-freeness is easily testable. Our techniques yield a few additional related results, but the problem of characterizing all easily testable graph properties, or even that of formulating a plausible conjectured characterization, remains open.

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.