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This study aimed to identify possible factors influencing the early outcome after medial open wedge high tibial osteotomy (MOWHTO). A total of 87 MOWHTO cases with a minimum of 2-year follow-up and second-look arthroscopic results available were enrolled. The cartilage degeneration state was evaluated by the International Cartilage Repair Society (ICRS) grading. Radiographic parameters including the hip–knee–ankle axis (HKA), medial proximal tibia angle (MPTA), posterior tibial slope, patellar height, mechanical lateral distal femoral angle (LDFA), joint line convergence angle (JLCA), joint line obliquity (JLO), and weight-bearing line ratio (WBLR) were measured. The pre-operative arthritic change was evaluated by Kellgren–Lawrence (KL) classification. According to the post-operative HKA, knees were divided into three (Under/Optimal/Over-correction) subgroups. Subjective International Knee Documentation Committee (IKDC) scores were evaluated and factors related to post-operative IKDC scores were analyzed. The pre-operative HKA (P = 0.002), post-operative HKA (P = 0.007), pre-operative MPTA (P = 0.011), and pre-operative WBLR (P = 0.031) were significantly related to the post-operative IKDC score. Cartilage degeneration states evaluated from first and second-look arthroscopy were not associated with post-operative IKDC score. Subgroup analysis revealed that the Under-correction group had significantly lower post-operative IKDC scores compared to the Optimal and Over-correction group (P = 0.012 and P = 0.030, respectively). Our result suggests that a sub-optimal correction of coronal limb alignment negatively affects the early outcome of MOWHTO. On the other hand, the effect of the degree of cartilage degeneration was not significant.

Lateral malleolus fracture is one of the most common fractures. However, there is controversy regarding the rehabilitation protocols used after surgery. In particular, the initiation point for weight-bearing has not been standardized. In the present study, we investigated the prognostic difference between immediate and delayed weight-bearing on lateral malleolus fractures. The medical records of matched patients in the immediate and delayed weight-bearing groups (50 and 50, respectively) were reviewed retrospectively. All patients were treated with open reduction and internal fixation using an anatomical locking compression plate with a lag screw. In the immediate weight-bearing group (IWB), tolerable weight-bearing (i.e., what can be endured immediately after surgery with crutches) was permitted. In the delayed weight-bearing group (DWB), weight-bearing was completely restricted for 4 weeks after surgery. Ankle motion exercise was permitted in both groups, starting from the day after surgery. Radiographic assessment data and clinical outcomes were reviewed between the two groups. No significant differences in radiographic assessments and complications were found between the two groups. Significant differences in terms of a shortened length of hospital stay and time to return to work with the IWB rehabilitation protocol compared with DWB were confirmed (6.0 vs. 9.2 days, p = 0.02 and 6.1 vs. 8.3 weeks, p = 0.02, respectively). A significant difference in sport factor was observed in the Foot and Ankle Outcome Score at 3 months postoperatively (72.3 vs. 67.4, p = 0.02). We found no significant differences between the two groups concerning postoperative radiological outcomes and complications. The benefits of shortening the time to return to work and length of hospital stay associated with the IWB rehabilitation protocol were confirmed. In conclusion, immediate weight-bearing is recommended in patients with lateral malleolus fracture after anatomical reduction and firm fixation by surgery.

Background:Few clinical studies have reported the predictors of lateral hinge fracture (LHF) after medial opening-wedge high tibial osteotomy (MOWHTO).Purpose/Hypothesis:The purpose was to compare the incidence of LHF on plain radiographs versus computed tomography (CT) scans and to investigate the factors related to the development of LHF after MOWHTO. We hypothesized that (1) a higher LHF detection rate would be seen on CT scans versus plain radiographs and (2) LHF incidence would be related to opening gap width and hinge position.Study Design:Cohort study (diagnosis); Level of evidence, 3.Methods:A total of 97 MOWHTO cases were included. The presence and types of LHF were determined from plain radiographs and CT scans. Radiographic parameters were measured on plain radiographs taken 6 weeks postoperatively. Anterior and posterior opening gap widths, coronal and sagittal osteotomy slopes, and fibular height were calculated from CT scans. The wedge-hinge relationship and the zone of hinge position were assessed, and the patient and radiographic factors related to LHF occurrence were evaluated.Results:Seventeen LHF cases (20.5%) were detected on plain radiographs, while significantly more (37 cases; 44.6%) were found on CT scans ( P = .001). Based on Takeuchi classification, 28 LHF cases were considered type 1, 7 were type 2, and 2 were type 3. Logistic regression analysis revealed that opening gap width (odds ratio, 1.615; 95% confidence interval, 1.232-2.118; P = .001) and posterior opening gap width (odds ratio, 3.731; 95% confidence interval, 1.642-4.351; P = .008,) differed significantly between patients with versus without LHF. Other patient and radiographic factors were not significantly related to LHF occurrence. Receiver operating characteristic curve analysis identified the opening gap width cutoff values for LHF as 11.0 mm (area under the curve, 0.81; sensitivity, 78.4%; specificity, 73.9%).Conclusion:The incidence of LHF after MOWHTO can be underestimated on plain radiographs compared with CT scans. Only large opening gap width, especially posterior gap width, was found to have a statistically significant relationship with occurrence of LHF. Therefore, special caution for possible LHF may be needed if a large correction is planned.

Background: Ankle joint swelling (AJS) in fracture patients is important for predicting soft tissue complications and deciding urgent surgery. Most surgeons still predict the progression of AJS based on their extensive experience. This study was conducted to identify factors that may affect the progression of AJS.Methods: The clinical data and radiographic images of 120 patients who have undergone surgical treatment for ankle fracture were obtained. The immediate radiographs after surgery were representative of the swelling of the ankle joint, and the radiographs 2 weeks after device removal were representative of the normal ankle joint. We defined AJS as the sum of the ankle joint skin contour diameters in the anteroposterior and lateral views. We performed multivariate regression analysis to compare the correlation between variables and AJS.Results: A total of 90 patients were included and followed up for an average of 18.1 months (range, 14–20). Three significant factors (preoperative serum glucose level, injury mechanism, and duration of surgery) were identified to affect AJS.Conclusion: Elevated preoperative serum glucose level, high-energy injury, and extended duration of surgery were more likely to aggravate AJS. These findings will help surgeons to predict the prognosis of patients with AJS and to provide proper management for patients with an ankle fracture.Level of Evidence: Retrospective cohort study, III

A set $S$ of natural numbers is a Sidon set if all the sums $s_1+s_2$ with $s_1$, $s_2\in S$ and $s_1\leq s_2$ are distinct. Let constants $\alpha>0$ and $0

Introduction Lateral malleolus fracture is one of the most common fractures. However, there is controversy regarding the rehabilitation protocols used after surgery. In particular, the initiation point for weight bearing has not been standardized. In the present study, we investigated the prognostic difference between immediate and delayed weight bearing on lateral malleolus fractures. Materials and Methods The medical records of matched 50 and 41 patients in the immediate and delayed weight-bearing groups were reviewed retrospectively. All patients were treated with open reduction and internal fixation using an anatomical locking compression plate. In the immediate weight-bearing group (IWB), tolerable weight bearing (i.e., can be endured immediately after surgery with crutches) was permitted. In the delayed weight-bearing group (DWB), weight bearing was completely restricted for 4 weeks after surgery. Ankle motion exercise was permitted in both groups starting from the day after surgery. Radiographic assessment data and clinical outcomes were reviewed. Results No significant differences in the radiographic assessments and complications were found between the two groups. Significant differences in shortened length of hospital stay and time to return to work with the IWB rehabilitation protocol were confirmed (IWB group vs. DWB group: 5.7 vs. 8.0 days and 6.0 vs. 8.0 months, respectively). A significant difference in sport factor was observed in the Foot and Ankle Outcome Score at 3 months postoperatively (75.5 vs 68.5). Conclusions We found no significant differences between the two groups with respect to postoperative radiological outcome and complications. The benefits of shortening the time to return to work and length of hospital stay associated with the IWB rehabilitation protocol were confirmed. In conclusion, immediate weight bearing is recommended in patients with lateral malleolus fracture after anatomical reduction and firm fixation by surgery.

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, which is itself a well known extension of the t -intersecting Erdźs-Ko-Rado theorem. The complete intersection theorem says that the maximum size of a family of subsets of n = { 1 , ź , n } , every pair of which intersects in at least t elements, is the size of certain trivially intersecting families proposed by Frankl. We address a cross intersecting version of their diametric theorem.Two families A and B of subsets of n are cross t -intersecting if for every A ź A and B ź B , A and B intersect in at least t elements. The p -weight of a k element subset A of n is p k ( 1 - p ) n - k , and the weight of a family A is the sum of the weights of its sets. The weight of a pair of families is the product of the weights of the families.The maximum p -weight of a t -intersecting family depends on the value of p . Ahlswede and Khachatrian showed that for p in the range r t + 2 r - 1 , r + 1 t + 2 r + 1 , the maximum p -weight of a t -intersecting family is that of the family F r t consisting of all subsets of n containing at least t + r elements of the set t + 2 r .In a previous paper we showed a cross t -intersecting version of this for large t in the case that r = 0 . In this paper, we do the same in the case that r = 1 . We show that for p in the range 1 t + 1 , 2 t + 3 the maximum p -weight of a cross t -intersecting pair of families, for t ź 200 , is achieved when both families are F 1 t . Further, we show that except at the endpoints of this range, this is, up to isomorphism, the only pair of t -intersecting families achieving this weight.

A set S of integers is a B 3 -set if all the sums of the form a 1 + a 2 + a 3 , with a 1 , a 2 and a 3 ? S and a 1 ? a 2 ? a 3 , are distinct. We obtain asymptotic bounds for the number of B 3 -sets of a given cardinality contained in the interval n = { 1 , ? , n } . We use these results to estimate the maximum size of a B 3 -set contained in a typical (random) subset of n of a given cardinality. These results confirm conjectures recently put forward by the authors On the number of B h -sets, Combin. Probab. Comput. 25 (2016), no. 1, 108-127.

Category: Trauma; Ankle Introduction/Purpose: Isolated lateral malleolus fracture is one of the most common fractures. However, there is controversy regarding the rehabilitation protocols used after surgery. In particular, the initiation point for the weight bearing has not been standardized. In the present study, we investigated the prognostic difference between the immediate and delayed weight bearing on isolated lateral malleolus fractures. Methods: Retrospectively, the medical records of matched 50 and 41patients in the immediate and delayed weight-bearing groups were reviewed. All patients were treated with open reduction and internal fixation using an anatomical locking compression plate. In the immediate weight-bearing group (IWB), tolerable weight bearing (i.e., that can be endured immediately after surgery with crutches) was permitted. In the delayed weight-bearing group (DWB), weight bearing was completely restricted for 4 weeks after surgery. Ankle motion exercise were permitted in both groups since postoperative day. Radiographic assessment data and the clinical outcomes were reviewed. Results: There were no significant differences noted in the radiographic assessments and complications between the two groups. Significant differences of shortening the return to work and hospital stays of IWB rehabilitation protocol were confirmed(5.7 vs. 8.0 days, 6.0 vs. 8.0 days in the IBW and DWB groups, respectively). In FAOS, there was a significant difference in the sport factor evaluated at postoperative 3 months(75.5 vs 68.5, respectively). Conclusion: There were no significant differences between the two groups in the postoperative radiological outcome and complications. And benefits of shortening the return to work and hospital stays of IWB rehabilitation protocol were confirmed. Conclusively, immediate weight bearing is recommended in patients with isolated lateral malleolus fracture, following anatomical reduction and firm fixation through surgery.

A setAof positive integers is aBh-setif all the sums of the forma1+ . . . +ah, witha1,. . .,ah∈Aanda1⩽ . . . ⩽ah, are distinct. We provide asymptotic bounds for the number ofBh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context ofBh-sets. We also use these results to estimate the maximum size of aBh-sets contained in a typical (random) subset of [n] with a given cardinality.

Two families, ${\mathcal A}$ and ${\mathcal B}$, of subsets of $[n]$ are cross $t$-intersecting if for every $A \in {\mathcal A}$ and $B \in {\mathcal B}$, $A$ and $B$ intersect in at least $t$ elements. For a real number $p$ and a family ${\mathcal A}$ the product measure $\mu_p ({\mathcal A})$ is defined as the sum of $p^{|A|}(1-p)^{n-|A|}$ over all $A\in{\mathcal A}$. For every non-negative integer $r$, and for large enough $t$, we determine, for any $p$ satisfying $\frac r{t+2r-1}\leq p\leq\frac{r+1}{t+2r+1}$, the maximum possible value of $\mu_p ({\mathcal A})\mu_p ({\mathcal B})$ for cross $t$-intersecting families ${\mathcal A}$ and ${\mathcal B}$. In this paper we prove a stronger stability result which yields the above result., Comment: 24 pages, 4 figures

Background The treatment of radial nerve palsy caused by closed humeral shaft fracture is a matter of debate. The purpose of this study is to evaluate the outcome of early surgical management of radial nerve palsy in patients with unstable closed humeral shaft fractures and to determine whether patients with this injury should be surgically explored. Methods Medical records of 15 consecutive patients who had undergone open reduction and internal fixation for unstable closed humeral shaft fractures were reviewed. Injury mechanisms, radiographs, intraoperative findings of the injured nerve, and clinical improvement were analyzed. Of the 15 patients, two were injured during simple falls, two during baseball pitching, one by a fall from a height, one during skiing, and one by direct machine impaction during work. Fracture locations were middle humerus in six patients, middle-distal in eight patients, and middle-proximal in one patient. Fracture patterns were transverse fracture in six patients, butterfly segment fracture in four patients, oblique segment fracture in three patients, and spiral fracture in two patients. Results Eleven patients showed macroscopically intact nerves, and four had total nerve transection. All four patients with total nerve transection were injured in traffic accident. Of these four patients, one showed comminuted fracture with butterfly fragment on the middle-distal shaft and the other three showed displaced transverse fracture on the middle to middle-distal shaft area. Transected nerves were directly repaired in two patients. In the other two patients, humerus shortening and nerve grafting were performed because of mangled nerve endings. All patients who had intact nerves showed fully recovered function at the last follow-up. Conclusion For radial nerve palsy accompanied by unstable humeral shaft fracture, primary exploration of the radial nerve and open reduction and plate fixation of the fracture should be considered as a treatment of choice. High-energy trauma, fracture location at the middle-distal humerus, and simple transverse fracture or comminuted fracture with butterfly fragment seems to be risk factors for radial nerve transection.

Two families A and B , of k-subsets of an n-set, are cross t-intersecting if for every choice of subsets A ? A and B ? B we have | A ? B | ? t . We address the following conjectured cross t-intersecting version of the Erd?s-Ko-Rado theorem: For all n ? ( t + 1 ) ( k - t + 1 ) the maximum value of | A | | B | for two cross t-intersecting families A , B ? ( n k ) is ( n - t k - t ) 2 . We verify this for all t ? 14 except finitely many n and k for each fixed t. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a p-weight version of the problem, which comes from the product measure on the power set of an n-set.

A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. The dynamic chromatic number@g"d(G) of a graph G is the least number k such that G has a dynamic coloring with k colors. We show that @g"d(G)@?4 for every planar graph except C"5, which was conjectured in Chen et al. (2012) [5]. The list dynamic chromatic numberch"d(G) of G is the least number k such that for any assignment of k-element lists to the vertices of G, there is a dynamic coloring of G where the color on each vertex is chosen from its list. Based on Thomassen's (1994) result [14] that every planar graph is 5-choosable, an interesting question is whether the list dynamic chromatic number of every planar graph is at most 5 or not. We answer this question by showing that ch"d(G)@?5 for every planar graph.

A set A of non-negative integers is called a Sidon set if all the sums , with and a1, , are distinct. A well-known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erdős, Singer and Turan from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non-differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erdős (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karo\'nski, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set $M$ of size $m$ where each element of $M$ belongs to each random subset with probability $p$, independently of all other elements in $M$. Fill, Scheinerman and Singer-Cohen [Random Struct Algorithms 16 (2000), 156--176] showed that the total variation distance between the random graph $G(n,m;p)$ and the Erd\"os-R\'enyi graph $G(n,\hat{p})$ tends to $0$ for any $0 \leq p=p(n) \leq 1$ if $m=n^{\alpha}$, $\alpha >6$, where $\hat{p}$ is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to $0$ for any $0 \leq p=p(n) \leq 1$ whenever $m \gg n^4$., Comment: revised version of the 1st draft "On a phase transition of the random intersection graph: Supercritical region"

For positive integers $d$ and $n$, let $[n]^d$ be the set of all vectors $(a_1,a_2,\dots, a_d)$, where $a_i$ is an integer with $0\leq a_i\leq n-1$. A subset $S$ of $[n]^d$ is called a \emph{Sidon set} if all sums of two (not necessarily distinct) vectors in $S$ are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $[n]^d$. First, let $\mathcal{Z}_{n,d}$ be the number of all Sidon sets in $[n]^d$. We show that $\log (\mathcal{Z}_{n,d})=\Theta(n^{d/2})$, where the constants of $\Theta$ depend only on $d$. Next, we estimate the maximum size of Sidon sets contained in a random set $[n]^d_p$, where $[n]^d_p$ denotes a random set obtained from $[n]^d$ by choosing each element independently with probability $p$., Comment: 14 pages, 1 figure, title was modified

For a family $\mathcal{F}$ of graphs, a graph $G$ is called \emph{$\mathcal{F}$-universal} if $G$ contains every graph in $\mathcal{F}$ as a subgraph. Let $\mathcal{F}_n(d)$ be the family of all graphs on $n$ vertices with maximum degree at most $d$. Dellamonica, Kohayakawa, R\"odl and Ruci\'nski showed that, for $d\geq 3$, the random graph $G(n,p)$ is $\mathcal{F}_n(d)$-universal with high probability provided $p\geq C\big(\frac{\log n}{n}\big)^{1/d}$ for a sufficiently large constant $C=C(d)$. In this paper we prove the missing part of the result, that is, the random graph $G(n,p)$ is $\mathcal{F}_n(2)$-universal with high probability provided $p\geq C\big(\frac{\log n}{n}\big)^{1/2}$ for a sufficiently large constant $C$., Comment: 13 pages, 1 figure

We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs., Rewriting with a major change (14 pages, 1 figure)

Study Design: A case report. Objectives: To report a rare case of extensive epidermoid cysts in the lumbosacral spine. Summary of Literature Review: The intradural epidermoid cyst with extensive involvement is rare, and previous reports have reported only extensive intramedullary epidermoid cysts. Materials and Methods: A 75-year-old male presented with progressive motor weakness of both extremities beginning 3 days prior. MRI showed extensive intradural extramedullary epidermoid cysts in the lumbosacral region. We performed total laminectomy from the L1 to the L5 level, and the cystic mass was removed. Results: We confirmed the epidermoid cyst on histopathologic examination. Conclusions: Extensive extramedullary epidermoid cysts are difficult to remove completely. Attempting complete removal may result in neurological deficit. Therefore, when surgical intervention is planned, the poor postoperative prognosis should be taken into consideration.

Purpose: The purpose of this study is to evaluate the usefulness of computed tomography (CT) for spiral tibia shaft fracture by analyzing associated distal tibia intra-articular fractures diagnosed by CT only which met the indication of surgical fixation and were fixed. Materials and Methods: Ninety-five spiral tibia shaft fractures with preoperative ankle plain radiographs and CT were analyzed retrospectively. The incidence and type of associated distal tibia articular fractures were evaluated by reviewing ankle plain radiography and CT. The number of fractures diagnosed by CT that correspond with the indication of fixation and that were actually fixed were analyzed. Results: Among 95 spiral tibia shaft fractures, 62 cases (65.3%) were associated with distal tibia intra-articular fracture. There were 37 cases of posterior malleolar fracture, 5 cases of avulsion fracture of the distal anterior tibiofibular ligament, 5 cases of medial malleolar fracture, and 15 cases of complex fracture. Among 52 posterior malleolar fractures including complex fracture, 20 cases were diagnosed by ankle plain radiograph. Of these 20 cases, 16 posterior malleolar fractures (80.0%) met the indication of surgical fixation, and 14 cases were actually fixed with a screw. Among 32 posterior malleolar fractures diagnosed by CT only, 26 cases (81.3%) met the indication of surgical fixation and 18 cases (56.3%) were fixed by screw. Conclusion: Approximately 50% of associated fractures were diagnosed by CT only and more than 80% of associated posterior malleolar fractures met the indication of surgical fixation and among these fractures, 18 cases (56.3%) were actually fixed by screw. This result suggests that CT is useful in diagnosis and treatment of distal tibia intra-articular fracture associated with spiral tibia shaft fracture.

Study Design: A case report. Objectives: To report a rare case of atypical spinal tuberculosis. Summary of Literature Review: In spinal tuberculosis, non-contiguous multifocal involvement and isolated involvement of posterior elements of the spine have been considered atypical features. There have been a few reports of each of these atypical features but no reports have described spinal tuberculosis with both of these atypical features. Materials and Methods: A 39-year-old man presented with back pain and progressive weakness of both lower extremities. He was diagnosed with spinal tuberculosis from the cervical to sacral spine, showing multifocal non-contiguous involvement with multiple abscesses on magnetic resonance imaging. Notably, in the thoracic spine area, isolated involvement of posterior elements was found with an epidural abscess compressing the spinal cord. He underwent a total laminectomy of the thoracic spine and multiple abscesses were drained with pigtail catheter insertions into the cervical, thoracic, and lumbar spine. Results: At the 8-month follow-up, the patient’s neurologic status had improved to Frankel Grade D, and the patient was able to walk with the support of a walker. At the 3-year follow-up, the patient had recovered completely without any neurologic deficit. Conclusions: Since atypical spinal tuberculosis may show various patterns, examination of the entire spine is important for early diagnosis. Treatment should be provided properly from minimally invasive procedures to open surgery depending on the extent of structural instability and neurologic deficit.